The Search command will return the sublist of the list indexlist consisting of all table entries for which the procedures in the procedurelists all return true.

If the table entries are Lie algebras, then the Search command will initialize each Lie algebra in indexlist and execute each procedure in the Liealgebraproperties list.  Each procedure in the Liealgebraproperties list will assume that a Lie algebra has been initialized.  No arguments may be used in defining these procedures.

If the table entries are Lie algebras of vector fields, then the procedures P in the argument VFSproperties = P should accept a single argument Gamma, this being the list of vector fields as provided by the table entry.  The Liealgebraproperties procedures are applied to the abstract Lie algebra defined by each table entry.  This abstract Lie algebra is automatically calculated and initialized by the Search command.

If the table entries are differential equations, then the procedures in DEproperties = P should accept a single argument called DE, this being the differential equation as provided by the table entry.  The VFSproperties procedures are applied to the symmetry algebra of the differential equation and the Liealgebraproperties are applied to the abstract Lie algebra defined by the symmetry algebra of the differential equation.  The symmetry algebra of DE will be automatically computed (using the PDEtools package) by the Search program, if required.

The command Search is part of the DifferentialGeometry:-Library package.  It can be used in the form Search(...) only after executing the commands with(DifferentialGeometry) and with(Library), but can always be used by executing DifferentialGeometry:-Library:-Search(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Library}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{P1}≔\left(\right)\→\mathrm{is}\left(\mathrm{Tools}:-\mathrm{DGinfo}\left(''LieAlgebraDimension''\right)\=4\right)$

$\mathrm{P1}≔\left(\right)\→\mathrm{is}\left(\mathrm{DifferentialGeometry}:-\mathrm{Tools}:-\mathrm{DGinfo}\left(''LieAlgebraDimension''\right)\=4\right)$

$\mathrm{P2}≔\left(\right)\mapsto \mathrm{Query}\left(''Solvable''\right)$

$\mathrm{P2}≔\left(\right)\→\mathrm{LieAlgebras}:-\mathrm{Query}\left(''Solvable''\right)$

$\mathrm{P3}≔\left(\right)\mapsto \mathbf{not}\mathrm{Query}\left(''Nilpotent''\right)$

$\mathrm{P3}≔\left(\right)\→\mathbf{not}\mathrm{LieAlgebras}:-\mathrm{Query}\left(''Nilpotent''\right)$

$\mathrm{Alglist1}≔\mathrm{Search}\left(''Winternitz''\,1\,\mathrm{Liealgebraproperties}=\left[\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\right]\right)$

$\mathrm{Alglist1}≔\left[\left[4\,2\right]\,\left[4\,3\right]\,\left[4\,4\right]\,\left[4\,5\right]\,\left[4\,6\right]\,\left[4\,7\right]\,\left[4\,8\right]\,\left[4\,9\right]\,\left[4\,10\right]\,\left[4\,11\right]\,\left[4\,12\right]\right]$

$\mathrm{P4}≔\left(\right)\mapsto \mathrm{is}\left(\mathrm{nops}\left(\mathrm{DerivedAlgebra}\left(\right)\right)=2\right)\:$

$\mathrm{Alglist2}≔\mathrm{Search}\left(''Winternitz''\,1\,\mathrm{indexlist}=\mathrm{Alglist1}\,\mathrm{Liealgebraproperties}=\left[\mathrm{P4}\right]\right)$

$\mathrm{Alglist2}≔\left[\left[4\,3\right]\,\left[4\,12\right]\right]$

$\mathrm{Browse}\left(''Winternitz''\,1\,\mathrm{Alglist2}\right)$

$''Winternitz''\,1\,\left[4\,3\right]$

$\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$''Winternitz''\,1\,\left[4\,12\right]$

$\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[y\right]\,M\,1\right)\:$

$\mathrm{Q1}≔\mathrm{Gamma}\mapsto \mathrm{is}\left(\mathrm{nops}\left(\mathrm{Gamma}\right)=3\right)$

$\mathrm{Q1}≔\mathrm{\Γ}\→\mathrm{is}\left(\mathrm{nops}\left(\mathrm{\Γ}\right)\=3\right)$

Q2 := proc(Gamma)

$\mathbf{local} \mathrm{\Γ1}\,A$

$\mathrm{\Γ1}≔\mathrm{map}\left(\mathrm{JetCalculus}:-\mathrm{Prolong}\,\mathrm{Gamma}\,1\right)$

$A≔\mathrm{GroupActions}:-\mathrm{IsotropySubalgebra}\left(\mathrm{\Γ1}\,\left[x=a\,y\left[\right]=b\,y\left[1\right]=c\right]\right)$

$\mathrm{is}\left(\mathrm{nops}\left(A\right)\ne 0\right)$

end:

$\mathrm{P1}≔\left(\right)\mapsto \mathrm{Query}\left(''Semisimple''\right)$

$\mathrm{P1}≔\left(\right)\→\mathrm{LieAlgebras}:-\mathrm{Query}\left(''Semisimple''\right)$

$\mathrm{VFS}≔\mathrm{Browse}\left(''Gonzalez-Lopez''\,1\right)$

$\mathrm{VFS}≔\left[\left[1\right]\,\left[2\right]\,\left[3\right]\,\left[4\right]\,\left[5\right]\,\left[6\right]\,\left[7\right]\,\left[8\right]\,\left[9\right]\,\left[10\right]\,\left[11\right]\,\left[12\right]\,\left[13\right]\,\left[14\right]\,\left[15\right]\,\left[16\right]\,\left[17\right]\,\left[18\right]\,\left[19\right]\,\left[20\,1\right]\,\left[20\,2\right]\,\left[20\,3\right]\,\left[20\,4\right]\,\left[20\,5\right]\,\left[21\,1\right]\,\left[21\,2\right]\,\left[21\,3\right]\,\left[21\,4\right]\,\left[21\,5\right]\,\left[22\,1\right]\,\left[22\,2\right]\,\left[22\,3\right]\,\left[22\,4\right]\,\left[22\,5\right]\,\left[22\,6\right]\,\left[22\,7\right]\,\left[22\,8\right]\,\left[22\,9\right]\,\left[22\,10\right]\,\left[22\,11\right]\,\left[22\,12\right]\,\left[22\,13\right]\,\left[22\,14\right]\,\left[22\,15\right]\,\left[22\,16\right]\,\left[22\,17\right]\,\left[22\,18\right]\,\left[22\,19\right]\,\left[22\,20\right]\,\left[22\,21\right]\,\left[22\,22\right]\,\left[22\,23\right]\,\left[22\,24\right]\,\left[22\,25\right]\,\left[22\,26\right]\,\left[22\,27\right]\,\left[22\,28\right]\,\left[22\,29\right]\,\left[22\,30\right]\,\left[22\,31\right]\,\left[22\,32\right]\,\left[22\,33\right]\,\left[22\,34\right]\,\left[22\,35\right]\,\left[22\,36\right]\,\left[22\,37\right]\,\left[23\,1\right]\,\left[23\,2\right]\,\left[23\,3\right]\,\left[23\,4\right]\,\left[23\,5\right]\,\left[23\,6\right]\,\left[23\,7\right]\,\left[23\,8\right]\,\left[23\,9\right]\,\left[23\,10\right]\,\left[23\,11\right]\,\left[23\,12\right]\,\left[23\,13\right]\,\left[23\,14\right]\,\left[23\,15\right]\,\left[23\,16\right]\,\left[23\,17\right]\,\left[23\,18\right]\,\left[23\,19\right]\,\left[23\,20\right]\,\left[23\,21\right]\,\left[23\,22\right]\,\left[23\,23\right]\,\left[23\,24\right]\,\left[23\,25\right]\,\left[23\,26\right]\,\left[23\,27\right]\,\left[23\,28\right]\,\left[23\,29\right]\,\left[23\,30\right]\,\left[23\,31\right]\,\left[23\,32\right]\,\left[23\,33\right]\,\left[23\,34\right]\,\left[23\,35\right]\,\left[23\,36\right]\,\left[23\,37\right]\,\left[24\,1\right]\,\left[24\,2\right]\,\left[24\,3\right]\,\left[24\,4\right]\,\left[24\,5\right]\,\left[25\,1\right]\,\left[25\,2\right]\,\left[25\,3\right]\,\left[25\,4\right]\,\left[25\,5\right]\,\left[26\,1\right]\,\left[26\,2\right]\,\left[26\,3\right]\,\left[26\,4\right]\,\left[26\,5\right]\,\left[27\,1\right]\,\left[27\,2\right]\,\left[27\,3\right]\,\left[27\,4\right]\,\left[27\,5\right]\,\left[28\,1\right]\,\left[28\,2\right]\,\left[28\,3\right]\,\left[28\,4\right]\,\left[28\,5\right]\right]$

$\mathrm{ans}≔\mathrm{Search}\left(''Gonzalez-Lopez''\,1\,\mathrm{indexlist}=\mathrm{VFS}\left[1..25\right]\,\mathrm{Liealgebraproperties}=\left[\mathrm{P1}\right]\,\mathrm{VFSproperties}=\left[\mathrm{Q1}\,\mathrm{Q2}\right]\,\mathrm{manifold}=M\right)$

$\mathrm{ans}≔\left[\left[11\right]\right]$

$\mathrm{Browse}\left(''Gonzalez-Lopez''\,1\,\mathrm{ans}\,\mathrm{manifold}=M\right)$

$''Gonzalez-Lopez''\,1\,\left[11\right]$

$\left[\mathrm{D\_x}\,x\mathrm{D\_x}\,x^2\mathrm{D\_x}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$