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Notebook[{
Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the point group of the Blasius equation \n\nYxxx+YYxx=0 \n\n\
discussed in section 8.10.1.",
  FontSize->14,
  FontVariations->{"CompatibilityType"->0}]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[StyleBox["First read in the package which is located in a \
folder called SymmetryAnalysis inside the Mathematica folder.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(Needs["\<SymmetryAnalysis`IntroToSymmetry`\>"]\)], "Input"],

Cell[TextData[StyleBox["Enter the input equation as a string. Don't include \
the ==0 at the end.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(blasiusequation = \n"\<y'''[x]+y[x]*y''[x]\>";\)\)], "Input"],

Cell[TextData[StyleBox["The function y[x] is a solution of the equation and \
this constraint must be applied in the form of a rule to the invariance \
condition. Be careful to check signs.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(rulesarray = \[IndentingNewLine]{"\<y'''[x]\[Rule]-y[x]*y''[x]\>"};\)\
\)], "Input"],

Cell[TextData[StyleBox["Enter the list of independent variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(independentvariables = {"\<x\>"};\)\)], "Input"],

Cell[TextData[StyleBox["Enter the list of dependent variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(dependentvariables = {"\<y\>"};\)\)], "Input"],

Cell[TextData[StyleBox["Enter the list of function names and constant names \
that need to be preserved when the equation is changed to (x1, y1) variables. \
In this case there are no such quantities and so we enter a null list.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(frozennames = {"\<\>"};\)\)], "Input"],

Cell[TextData[StyleBox["Enter the maximum derivative order.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(p = 3;\)\)], "Input"],

Cell[TextData[StyleBox["The maximum derivative order that the infinitesimals \
are assumed to depend on is specified by the input parameter r. This \
parameter is only nonzero when the user is looking for Lie contact groups or \
Lie-Backlund groups. For the usual case where one is searching for point \
groups set r=0.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(r = 0;\)\)], "Input"],

Cell[TextData[StyleBox["When searching for Lie-B\[ADoubleDot]cklund groups \
(r=1 or greater) one can, without loss of generality, leave the independent \
variables untransformed. The corresponding infinitesimals (the xse's) are set \
to zero by setting xseon=0. If one is searching for point groups then set \
xseon=1. The choice xseon=1 is also an option when looking for Lie-B\
\[ADoubleDot]cklund groups and this can be useful when looking for contact \
symmetries.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(xseon = 1;\)\)], "Input"],

Cell[TextData[StyleBox["When searching for Lie-B\[ADoubleDot]cklund groups it \
is necessary to differentiate the input equation r times producing \
derivatives of order p+r. These higher order differential consequences are \
appended to the set of rules applied to the invariance condition. This \
process is carried out automatically when internalrules=1. For point groups \
the equation or equation system is the only rule or set of rules needed and \
one sets internalrules=0.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(\(internalrules = 0;\)\)], "Input"],

Cell[TextData[StyleBox["Now work out the determining equations of the Lie \
point group that leaves the equation  invariant. The output is available as a \
table of strings called zdeterminingequations.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[
      FindDeterminingEquations[\[IndentingNewLine]independentvariables, 
        dependentvariables, frozennames, p, r, xseon, blasiusequation, 
        rulesarray, internalrules]]\)], "Input"],

Cell[BoxData[
    \("FindDeterminingEquations has finished executing. You can look at the \
output in the table zdeterminingequations. Each entry in this table is a \
determining equation in string format expressed in terms of z-variables. \
Rules for converting between z-variables and conventional variables are \
contained in the table ztableofrules. To view the determining equations in \
terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy."\)], "Print"],

Cell[BoxData[
    \({0.7500000000000001`\ Second, Null}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(equationgenericvariables\)], "Input"],

Cell[BoxData[
    \("y1'''[x1]+y1[x1]*y1''[x1]"\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(invarconditiontable\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{"0", ",", 
        RowBox[{
          SuperscriptBox["y1", "\[Prime]\[Prime]",
            MultilineFunction->None], "[", "x1", "]"}], ",", "0", 
        ",", \(y1[x1]\), ",", "1"}], "}"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["The program expresses the determining equations in \
terms of z-variables. Here is the correspondence between z-variables and \
conventional variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(ztableofrules\)], "Input"],

Cell[BoxData[
    \({z1 \[Rule] x, z2 \[Rule] y[x]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Here are the determining equations expressed in terms \
of z-variables. This output may be quite long. The equations in the table can \
be distinguished by the ==0 at the end of each item.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(zdeterminingequations\)], "Input"],

Cell[BoxData[
    \({"eta1[z1, z2] + z2*Derivative[1, 0][xse1][z1, z2] + 3*Derivative[1, \
1][eta1][z1, z2] - 3*Derivative[2, 0][xse1][z1, z2] == 0", 
      "z2*Derivative[0, 1][xse1][z1, z2] + 3*Derivative[0, 2][eta1][z1, z2] - \
9*Derivative[1, 1][xse1][z1, z2] == 0", 
      "-3*Derivative[0, 1][xse1][z1, z2] == 0", 
      "z2*Derivative[0, 2][eta1][z1, z2] - 2*z2*Derivative[1, 1][xse1][z1, \
z2] + 3*Derivative[1, 2][eta1][z1, z2] - 3*Derivative[2, 1][xse1][z1, z2] == \
0", "-(z2*Derivative[0, 2][xse1][z1, z2]) + Derivative[0, 3][eta1][z1, z2] - \
3*Derivative[1, 2][xse1][z1, z2] == 0", 
      "-6*Derivative[0, 2][xse1][z1, z2] == 0", 
      "-Derivative[0, 3][xse1][z1, z2] == 0", 
      "2*z2*Derivative[1, 1][eta1][z1, z2] - z2*Derivative[2, 0][xse1][z1, \
z2] + 3*Derivative[2, 1][eta1][z1, z2] - Derivative[3, 0][xse1][z1, z2] == \
0", "z2*Derivative[2, 0][eta1][z1, z2] + Derivative[3, 0][eta1][z1, z2] == \
0"}\)], "Output"]
}, Open  ]],

Cell["\<\
Lets look at the determining equations in a slightly more readable \
form.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(zdeterminingequationscolumn = \[IndentingNewLine]TraditionalForm[\
\[IndentingNewLine]ColumnForm[ToExpression[zdeterminingequations], Left, 
          Below]]\)], "Input"],

Cell[BoxData[
    FormBox[
      InterpretationBox[GridBox[{
            {
              RowBox[{
                RowBox[{\(eta1(z1, z2)\), "+", 
                  RowBox[{"z2", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((1, 0)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((1, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((2, 0)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}]}], 
                "==", "0"}]},
            {
              RowBox[{
                RowBox[{
                  RowBox[{"z2", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((0, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((0, 2)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{"9", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((1, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}]}], 
                "==", "0"}]},
            {
              RowBox[{
                RowBox[{\(-3\), " ", 
                  RowBox[{
                    SuperscriptBox["xse1", 
                      TagBox[\((0, 1)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}]}], "==",
                 "0"}]},
            {
              RowBox[{
                RowBox[{
                  RowBox[{"z2", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((0, 2)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{"2", " ", "z2", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((1, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((1, 2)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((2, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}]}], 
                "==", "0"}]},
            {
              RowBox[{
                RowBox[{
                  RowBox[{\(-z2\), " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((0, 2)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{
                    SuperscriptBox["eta1", 
                      TagBox[\((0, 3)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}], "-", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((1, 2)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}]}], 
                "==", "0"}]},
            {
              RowBox[{
                RowBox[{\(-6\), " ", 
                  RowBox[{
                    SuperscriptBox["xse1", 
                      TagBox[\((0, 2)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}]}], "==",
                 "0"}]},
            {
              RowBox[{
                RowBox[{"-", 
                  RowBox[{
                    SuperscriptBox["xse1", 
                      TagBox[\((0, 3)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}]}], "==",
                 "0"}]},
            {
              RowBox[{
                RowBox[{
                  RowBox[{"2", " ", "z2", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((1, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{"z2", " ", 
                    RowBox[{
                      SuperscriptBox["xse1", 
                        TagBox[\((2, 0)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{"3", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((2, 1)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "-", 
                  RowBox[{
                    SuperscriptBox["xse1", 
                      TagBox[\((3, 0)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}]}], "==",
                 "0"}]},
            {
              RowBox[{
                RowBox[{
                  RowBox[{"z2", " ", 
                    RowBox[{
                      SuperscriptBox["eta1", 
                        TagBox[\((2, 0)\),
                          Derivative],
                        MultilineFunction->None], "(", \(z1, z2\), ")"}]}], 
                  "+", 
                  RowBox[{
                    SuperscriptBox["eta1", 
                      TagBox[\((3, 0)\),
                        Derivative],
                      MultilineFunction->None], "(", \(z1, z2\), ")"}]}], "==",
                 "0"}]}
            },
          GridBaseline->Top,
          ColumnAlignments->{Left}],
        ColumnForm[ {
          Equal[ 
            Plus[ 
              eta1[ z1, z2], 
              Times[ z2, 
                Derivative[ 1, 0][ xse1][ z1, z2]], 
              Times[ 3, 
                Derivative[ 1, 1][ eta1][ z1, z2]], 
              Times[ -3, 
                Derivative[ 2, 0][ xse1][ z1, z2]]], 0], 
          Equal[ 
            Plus[ 
              Times[ z2, 
                Derivative[ 0, 1][ xse1][ z1, z2]], 
              Times[ 3, 
                Derivative[ 0, 2][ eta1][ z1, z2]], 
              Times[ -9, 
                Derivative[ 1, 1][ xse1][ z1, z2]]], 0], 
          Equal[ 
            Times[ -3, 
              Derivative[ 0, 1][ xse1][ z1, z2]], 0], 
          Equal[ 
            Plus[ 
              Times[ z2, 
                Derivative[ 0, 2][ eta1][ z1, z2]], 
              Times[ -2, z2, 
                Derivative[ 1, 1][ xse1][ z1, z2]], 
              Times[ 3, 
                Derivative[ 1, 2][ eta1][ z1, z2]], 
              Times[ -3, 
                Derivative[ 2, 1][ xse1][ z1, z2]]], 0], 
          Equal[ 
            Plus[ 
              Times[ -1, z2, 
                Derivative[ 0, 2][ xse1][ z1, z2]], 
              Derivative[ 0, 3][ eta1][ z1, z2], 
              Times[ -3, 
                Derivative[ 1, 2][ xse1][ z1, z2]]], 0], 
          Equal[ 
            Times[ -6, 
              Derivative[ 0, 2][ xse1][ z1, z2]], 0], 
          Equal[ 
            Times[ -1, 
              Derivative[ 0, 3][ xse1][ z1, z2]], 0], 
          Equal[ 
            Plus[ 
              Times[ 2, z2, 
                Derivative[ 1, 1][ eta1][ z1, z2]], 
              Times[ -1, z2, 
                Derivative[ 2, 0][ xse1][ z1, z2]], 
              Times[ 3, 
                Derivative[ 2, 1][ eta1][ z1, z2]], 
              Times[ -1, 
                Derivative[ 3, 0][ xse1][ z1, z2]]], 0], 
          Equal[ 
            Plus[ 
              Times[ z2, 
                Derivative[ 2, 0][ eta1][ z1, z2]], 
              Derivative[ 3, 0][ eta1][ z1, z2]], 0]}, Left, Below],
        Editable->False], TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
Now solve the determining equations in terms of multivariable \
polynomials of a selected order. The Mathematica function Solve uses Gaussian \
elimination to solve a large number of linear equations for the polynomial \
coefficients. The time roughly follows

time/timeref=((number of equations)/(number of equationsref))^n

where the exponent is between 2.4 and 2.7. The Mathematica function Timing \
outputs the time required for the SolveDeterminingEquations function to \
execute.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Timing[
      SolveDeterminingEquations[independentvariables, dependentvariables, r, 
        xseon, zdeterminingequations, order = 3]]\)], "Input"],

Cell[BoxData[
    InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(20\\)\"\>",
      StringForm[ "The number of unknown polynomial coefficients = ``", 20],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(43\\)\"\>",
      StringForm[ 
      "The number of equations for the polynomial coefficients = ``", 43],
      Editable->False]], "Print"],

Cell[BoxData[
    \("SolveDeterminingEquations has finished executing. You can look at the \
output in the tables xsefunctions and etafunctions. Each entry in these \
tables is an infinitesimal function in string format expressed in terms of \
z-variables and the group parameters. The output can also be viewed with the \
group parameters stripped away by looking at the table infinitesimalgroups. \
In either case you may wish to convert the z-variables to conventional \
variables using the table ztableofrules.   Keep in mind that this function \
only finds solutions of the determining equations that are of algebraic form. \
The determining equations may admit solutions that involve transcendental \
functions and/or integrals. Note that arbitrary functions may appear in the \
infinitesimals and that these can be detected by running the package function \
SolveDeterminingEquations for several polynomial orders. If terms of ever \
increasing order appear, then an arbitrary function is indicated."\)], "Print"],

Cell[BoxData[
    \({0.18333333333333335`\ Second, Null}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Here are the infinitesimals for the independent \
variables expressed in terms of z-variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(xsefunctions\)], "Input"],

Cell[BoxData[
    \({"xse1[z1_, z2_]=a10 - b14*z1"}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["and the infinitesimals for the dependent variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(etafunctions\)], "Input"],

Cell[BoxData[
    \({"eta1[z1_, z2_]=b14*z2"}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Express the xse functions in terms of x,y variables.",
  
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(ToExpression[xsefunctions] /. ztableofrules\)], "Input"],

Cell[BoxData[
    \({a10 - b14\ x}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Express the eta function in terms of these \
variables.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
    \(ToExpression[etafunctions] /. ztableofrules\)], "Input"],

Cell[BoxData[
    \({b14\ y[x]}\)], "Output"]
}, Open  ]],

Cell["Here is a list of the various infinitesimal groups.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(infinitesimalgroupsxy = 
      infinitesimalgroups /. {z1 \[Rule] x, z2 \[Rule] y}\)], "Input"],

Cell[BoxData[
    \({{{1}, {0}}, {{\(-x\)}, {y}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ColumnForm[infinitesimalgroupsxy]\)], "Input"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {\({{1}, {0}}\)},
          {\({{\(-x\)}, {y}}\)}
          },
        GridBaseline->{Baseline, {1, 1}},
        ColumnAlignments->{Left}],
      ColumnForm[ {{{1}, {0}}, {{
        Times[ -1, x]}, {y}}}],
      Editable->False]], "Output"]
}, Open  ]],

Cell[TextData[StyleBox["The final result indicates that the equation is \
invariant under two groups. The parameter a10 corresponds to a translation in \
x and the parameter b14 represents a dilation in x and y. Generate the \
commutator table of the Lie algebra. ",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
    \(MakeCommutatorTable[independentvariables, dependentvariables, 
      infinitesimalgroupsxy]\)], "Input"],

Cell[BoxData[
    \("MakeCommutatorTable has finished executing. You can look at the output \
in the table commutatortable. To present the output in the most readable form \
you may want view it as a matrix using MatrixForm[commutatortable]. \
Occasionally the entries in the commutatortable will have terms that cancel. \
To get rid of these terms use the function Simplify before viewing the \
table."\)], "Print"]
}, Open  ]],

Cell[TextData[StyleBox["Put the commutator table in readable form.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell[BoxData[
    \(MatrixForm[commutatortable]\)], "Input"],

Cell[BoxData[
    InterpretationBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {
              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {"0"}
                    }], "\[NoBreak]", ")"}], 
              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {\(-1\)},
                    {"0"}
                    }], "\[NoBreak]", ")"}]},
            {
              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"1"},
                    {"0"}
                    }], "\[NoBreak]", ")"}], 
              RowBox[{"(", "\[NoBreak]", GridBox[{
                    {"0"},
                    {"0"}
                    }], "\[NoBreak]", ")"}]}
            }], "\[NoBreak]", ")"}],
      MatrixForm[ {{{{0}, {0}}, {{-1}, {0}}}, {{{1}, {0}}, {{0}, {
        0}}}}]]], "Output"]
}, Open  ]],

Cell[TextData[StyleBox["This is a solvable Lie algebra that can be used to \
reduce the Blasius equation to first order.",
  FontWeight->"Plain"]], "Text",
  ImageRegion->{{0, 1}, {0, 1}}]
},
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