(* Content-type: application/mathematica *)

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(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the point group of the Boussinesq PDE, \n\nfxxx+fx^2+f fxx+ftt=0.",
 FontSize->14,
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Clear all symbols in the current context.", "Text",
 CellChangeTimes->{{3.4715644485587473`*^9, 3.47156448605101*^9}}],

Cell[BoxData[
 RowBox[{"ClearAll", "[", 
  RowBox[{"Evaluate", "[", 
   RowBox[{
    RowBox[{"Context", "[", "]"}], "<>", "\"\<*\>\""}], "]"}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
  FontWeight->"Plain"],
 StyleBox["Mathematica",
  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
  FontWeight->"Plain"]
}], "Text",
 CellChangeTimes->{{3.472046215957556*^9, 3.472046255573135*^9}, {
  3.472046289795615*^9, 3.472046308115048*^9}, {3.4720466725423717`*^9, 
  3.472046673357225*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{"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[
 RowBox[{
  RowBox[{
  "boussinesqequation", "=", "\n", 
   "\"\<D[f[x,t],x,x,x,x]+D[f[x,t],x]^2+f[x,t]*D[f[x,t],x,x]-D[f[x,t],t,t]\>\"\
"}], ";"}]], "Input"],

Cell[TextData[{
 StyleBox["The function ",
  FontWeight->"Plain"],
 "f[x,t]",
 StyleBox[" 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[
 RowBox[{
  RowBox[{"rulesarray", "=", "\[IndentingNewLine]", 
   RowBox[{
   "{", "\"\<D[f[x,t],x,x,x,x]->-D[f[x,t],x]^2-f[x,t]*D[f[x,t],x,x]+D[f[x,t],\
t,t]\>\"", "}"}]}], ";"}]], "Input"],

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

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

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

Cell[BoxData[
 RowBox[{
  RowBox[{"dependentvariables", "=", 
   RowBox[{"{", "\"\<f\>\"", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of function and/or constant names that \
must be preserved when the equation is expressed in terms of generic \
y1[x1,x2] variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"frozennames", "=", 
   RowBox[{"{", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the maximum derivative order of the input \
equation(s).",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"p", "=", "4"}], ";"}]], "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[
 RowBox[{
  RowBox[{"r", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund 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-Backlund groups and \
this can be useful when looking for contact symmetries.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"xseon", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups it is \
necessary to differentiate the input equation with respect to each \
independent variable to produce all possible 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[
 RowBox[{
  RowBox[{"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[
 RowBox[{"FindDeterminingEquations", "[", "\[IndentingNewLine]", 
  RowBox[{
  "independentvariables", ",", "dependentvariables", ",", "frozennames", ",", 
   "p", ",", "r", ",", "xseon", ",", "\[IndentingNewLine]", 
   "boussinesqequation", ",", "rulesarray", ",", "internalrules"}], 
  "]"}]], "Input"],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(25090712\\), the time used = \\!\\(0.699093`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 25090712, 0.699093],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404035511957*^9, 3.471555084677692*^9, 
  3.4715787781026583`*^9, 3.483656416746492*^9, 3.48374498928045*^9}],

Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.469404035511957*^9, 3.471555084677692*^9, 
  3.4715787781026583`*^9, 3.483656416746492*^9, 3.483744989530429*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(44\\), the \
byte count of zdeterminingequations = \\!\\(5488\\), the memory in use = \
\\!\\(25619880\\), the time used = \\!\\(0.959587`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
= ``, the memory in use = ``, the time used = ``", 44, 5488, 25619880, 
   0.959587],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404035511957*^9, 3.471555084677692*^9, 
  3.4715787781026583`*^9, 3.483656416746492*^9, 3.483744989567993*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(25620936\\), the time used = \\!\\(0.959857`\\)\"\>",
  StringForm[
  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 25620936, 0.959857],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404035511957*^9, 3.471555084677692*^9, 
  3.4715787781026583`*^9, 3.483656416746492*^9, 3.4837449895691977`*^9}],

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",
 CellChangeTimes->{3.469404035511957*^9, 3.471555084677692*^9, 
  3.4715787781026583`*^9, 3.483656416746492*^9, 3.483744989571084*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData["equationgenericvariables"], "Input"],

Cell[BoxData["\<\"D[y1[x1,x2],x1,x1,x1,x1]+D[y1[x1,x2],x1]^2+y1[x1,x2]*D[y1[\
x1,x2],x1,x1]-D[y1[x1,x2],x2,x2]\"\>"], "Output",
 CellChangeTimes->{3.469404035823633*^9, 3.47155508508574*^9, 
  3.471578778493504*^9, 3.483656417083839*^9, 3.483744989623981*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData["invarconditiontable"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", "0", ",", 
   RowBox[{
    SuperscriptBox["y1", 
     TagBox[
      RowBox[{"(", 
       RowBox[{"2", ",", "0"}], ")"}],
      Derivative],
     MultilineFunction->None], "[", 
    RowBox[{"x1", ",", "x2"}], "]"}], ",", 
   RowBox[{"2", " ", 
    RowBox[{
     SuperscriptBox["y1", 
      TagBox[
       RowBox[{"(", 
        RowBox[{"1", ",", "0"}], ")"}],
       Derivative],
      MultilineFunction->None], "[", 
     RowBox[{"x1", ",", "x2"}], "]"}]}], ",", "0", ",", 
   RowBox[{"y1", "[", 
    RowBox[{"x1", ",", "x2"}], "]"}], ",", "0", ",", 
   RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "1", ",", 
   "0", ",", "0", ",", "0", ",", "0"}], "}"}]], "Output",
 CellChangeTimes->{3.4694040358865347`*^9, 3.471555085157504*^9, 
  3.471578778565619*^9, 3.48365641713875*^9, 3.483744989673133*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData["ztableofrules"], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"z1", "\[Rule]", "x"}], ",", 
   RowBox[{"z2", "\[Rule]", "t"}], ",", 
   RowBox[{"z3", "\[Rule]", 
    RowBox[{"f", "[", 
     RowBox[{"x", ",", "t"}], "]"}]}]}], "}"}]], "Output",
 CellChangeTimes->{3.469404035936071*^9, 3.4715550852407217`*^9, 
  3.471578778649077*^9, 3.483656417184301*^9, 3.4837449897058907`*^9}]
}, 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[
 RowBox[{"zdeterminingequations", ";"}]], "Input",
 CellChangeTimes->{3.471555100170415*^9}]
}, Open  ]],

Cell["How many determining equations are there?", "Text"],

Cell[CellGroupData[{

Cell["Length[zdeterminingequations]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData["44"], "Output",
 CellChangeTimes->{3.4694040360536137`*^9, 3.471555085418989*^9, 
  3.471578778769536*^9, 3.4836564172799664`*^9, 3.483744989782151*^9}]
}, Open  ]],

Cell["\<\
Now solve the determining equations in terms of multivariable polynomials of \
a selected order. This function can take a long time. 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[
 RowBox[{"Timing", "[", 
  RowBox[{"SolveDeterminingEquations", "[", "\[IndentingNewLine]", 
   RowBox[{
   "independentvariables", ",", "dependentvariables", ",", "r", ",", "xseon", 
    ",", "zdeterminingequations", ",", 
    RowBox[{"order", "=", "4"}]}], "]"}], "]"}]], "Input"],

Cell[BoxData[
 InterpretationBox["\<\"The variable powertablelength is the number of terms \
required for each multivariate polynomial used for the infinitesimals. This \
number is determined by the choice of polynomial order and the number of \
zvariables. The time needed to solve the determining equations increases as \
powertable increases. powertablelength = \\!\\(35\\)\"\>",
  StringForm[
  "The variable powertablelength is the number of terms required for each \
multivariate polynomial used for the infinitesimals. This number is \
determined by the choice of polynomial order and the number of zvariables. \
The time needed to solve the determining equations increases as powertable \
increases. powertablelength = ``", 35],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989832891*^9}],

Cell[BoxData["\<\"The polynomial expansions have been substituted into the \
determining equations. It is now time to collect the coefficients of various \
powers of zvariables into a table called table of coefficientsall. This step \
uses the function CoefficientList and is a fairly time consuming procedure.\"\
\>"], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989893396*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26285760\\), The time = \\!\
\\(1.0410430000000002`\\)\"\>",
  StringForm[
  "The memory in use = ``, The time = ``", 26285760, 1.0410430000000002`],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989894499*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of unknown polynomial coefficients = \
\\!\\(105\\)\"\>",
  StringForm["The number of unknown polynomial coefficients = ``", 105],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.4837449899403563`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The number of equations for the polynomial \
coefficients = \\!\\(446\\)\"\>",
  StringForm[
  "The number of equations for the polynomial coefficients = ``", 446],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.48374498994156*^9}],

Cell[BoxData["\<\"Now it we are ready to use the function Solve to find the \
nonzero polynomial coefficients corresponding to the symmetries of the input \
equation(s). This can take a while.\"\>"], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989942644*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26495872\\), The time = \\!\
\\(1.056184`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26495872, 1.056184],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989943947*^9}],

Cell[BoxData["\<\"Solve has finished.\"\>"], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989945032*^9}],

Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
executing.\"\>"], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989946067*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The memory in use = \\!\\(26696432\\), The time = \\!\
\\(1.068738`\\)\"\>",
  StringForm["The memory in use = ``, The time = ``", 26696432, 1.068738],
  Editable->False]], "Print",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.4837449899471684`*^9}],

Cell[BoxData["\<\"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. \
\\n\\nKeep in mind that this function only finds solutions of the determining \
equations that are of polynomial 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",
 CellChangeTimes->{3.469404036156178*^9, 3.471555085613628*^9, 
  3.4715787789533567`*^9, 3.483656417355012*^9, 3.483744989948224*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0.10473599999999994`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{3.4694040361929502`*^9, 3.471555085731352*^9, 
  3.471578779070713*^9, 3.4836564175087557`*^9, 3.483744989950254*^9}]
}, 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[
 RowBox[{"{", 
  RowBox[{"\<\"xse1[z1_, z2_, z3_]=a10 - (b115*z1)/2\"\>", 
   ",", "\<\"xse2[z1_, z2_, z3_]=a20 - b115*z2\"\>"}], "}"}]], "Output",
 CellChangeTimes->{3.469404036224859*^9, 3.471555085791396*^9, 
  3.471578779103175*^9, 3.483656417630685*^9, 3.4837449900645657`*^9}]
}, 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[
 RowBox[{"{", "\<\"eta1[z1_, z2_, z3_]=b115*z3\"\>", "}"}]], "Output",
 CellChangeTimes->{3.4694040363652153`*^9, 3.471555085872953*^9, 
  3.471578779160721*^9, 3.483656417690193*^9, 3.483744990123082*^9}]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"ToExpression", "[", "xsefunctions", "]"}], "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"z1", "\[Rule]", "x"}], ",", 
    RowBox[{"z2", "\[Rule]", "t"}], ",", 
    RowBox[{"z3", "\[Rule]", "f"}]}], "}"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"a10", "-", 
    FractionBox[
     RowBox[{"b115", " ", "x"}], "2"]}], ",", 
   RowBox[{"a20", "-", 
    RowBox[{"b115", " ", "t"}]}]}], "}"}]], "Output",
 CellChangeTimes->{3.469404036408354*^9, 3.471555085934617*^9, 
  3.4715787791921883`*^9, 3.4836564177395372`*^9, 3.4837449901562023`*^9}]
}, Open  ]]
}, 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[
 RowBox[{
  RowBox[{"ToExpression", "[", "etafunctions", "]"}], "/.", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"z1", "\[Rule]", "x"}], ",", 
    RowBox[{"z2", "\[Rule]", "t"}], ",", 
    RowBox[{"z3", "\[Rule]", "f"}]}], "}"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"b115", " ", "f"}], "}"}]], "Output",
 CellChangeTimes->{3.469404036453772*^9, 3.471555085987196*^9, 
  3.471578779242652*^9, 3.483656417786064*^9, 3.4837449902064123`*^9}]
}, Open  ]],

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

Cell[BoxData[
 RowBox[{
  RowBox[{"infinitesimalgroupsxy", "=", 
   RowBox[{"infinitesimalgroups", "/.", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"z1", "\[Rule]", "x"}], ",", 
      RowBox[{"z2", "\[Rule]", "t"}], ",", 
      RowBox[{"z3", "\[Rule]", "f"}]}], "}"}]}]}], ";"}]], "Input"],

Cell[BoxData[
 RowBox[{"Column", "[", "infinitesimalgroupsxy", "]"}]], "Input"],

Cell[BoxData[
 TagBox[GridBox[{
    {
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "0"}], "}"}], ",", 
       RowBox[{"{", "0", "}"}]}], "}"}]},
    {
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "1"}], "}"}], ",", 
       RowBox[{"{", "0", "}"}]}], "}"}]},
    {
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"-", 
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