Let $M$ be a manifold and let $\left[T_1\, T_2\, ..\.T_N\right]$ be a list of tensor fields on $M$. Then the Lie algebra $\mathrm{\Γ}$of infinitesimal symmetries of the list of tensors $T_i$ is the Lie algebra of vector fields $X$on $M$ such that the Lie derivatives ${\mathrm{\ℒ}}_X T_i \= 0$for $i \= 1\, 2\, ..\. \, N\.$  

If the tensors $T_i$all have the same tensorial type, say $T_i \in T_s^r\left(M\right)\,$then let $\mathrm{\𝒯}\mathit{ }\mathit{\=}\mathit{ }$span$\left\{T_1\, T_2\, ..\.T_N\right\}$. Then the Lie algebra $\mathrm{\Γ}$of infinitesimal symmetries of the tensor space $\mathrm{\𝒯}\mathit{ }$is Lie algebra of vector fields $X$on $M$ such that $$${\mathrm{\ℒ}}_X T_i \in \mathrm{\𝒯}\mathit{ }$for $i \= 1\, 2\, ..\. \, N\.$

The command InfinitesimalSymmetriesOfGeometricObjectFields(T) calculates the Lie algebra of infinitesimal symmetries of the tensors and tensor spaces in the list T. For example, if$$$T_1\, T_2\, T_3\,$$T_4$ are 4 tensor fields and T$$$\= \left[T_1\, T_2\, \left[T_3\,T_4\right]\right]$, then InfinitesimalSymmetriesOfGeometricObjectFields(T) will return the Lie algebra of vector fields $X$such that ${\mathrm{ℒ}}_X T_1 \= 0 \, {\mathrm{ℒ}}_X T_1 \= 0 \, {\mathrm{\ℒ}}_X T_3 \in$span $\left\{T_3\,T_4\right\}\,$${\mathrm{ℒ}}_X T_4 \in$span $\left\{T_3\,T_4\right\}$.  

The procedure InfinitesimalSymmetriesOfGeometricObjectFields creates an arbitrary vector field $X$on $M$and generates a system of first order PDE for the coefficients of $X$from the Lie derivative equations ${\mathrm{ℒ}}_X T_i \= 0$$$and ${\mathrm{\ℒ}}_X T_i \in \mathrm{\𝒯}$. These PDE are solved using pdsolve .

If the (real) Lie algebra $\mathrm{\Γ}$ of infinitesimal symmetries for a given collection of geometric object fields is finite dimensional (so that the most general infinitesimal symmetry depends only upon arbitrary constants), then the optional argument output = "list" will return a basis for $\mathrm{\Γ}$.

With the option output = "pde", just the determining differential equations for the symmetries are returned.

The variables appearing in the coefficients of the vector field X can be specified with the option coefficientvariables = [x1, x2, ...].

The exact form of the infinitesimal symmetries to be found can be specified with the option ansatz = X. With this option, the unknown coefficients to be solved for must be explicitly identified with the option unknowns = [F1, F2, ...].

Additional constraints on the symmetry vector field X can be specified with the optional argument auxiliaryequations = [Delta1, Delta2,..], where Delta1, Delta2,.. are differential equations whose unknowns are the coefficients of the vector field X.

If the given geometric object fields T depend upon parameters {a1, a2, ...}, then the optional argument parameters = {a1, a2, ...} will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of infinitesimal symmetries, one list for each set of parameter values, will be returned.

Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.

If pdsolve is unable to explicitly solve the pde system for the infinitesimal symmetries, then NULL is returned.

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

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$

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

$\mathrm{frame\; name:\; M}$

$Y≔\mathrm{D\_x}$

$Y:=\mathrm{D\_x}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[Y\right]\right)$

$\mathrm{\_F3}\left(y\,z\right)\mathrm{D\_x}\+\mathrm{\_F2}\left(y\,z\right)\mathrm{D\_y}\+\mathrm{\_F1}\left(y\,z\right)\mathrm{D\_z}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[Y\right]\,\mathrm{coefficientvariables}=\left[x\right]\right)$

$\mathrm{D\_x}\mathrm{\_C3}\+\mathrm{D\_y}\mathrm{\_C2}\+\mathrm{D\_z}\mathrm{\_C1}$

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right]\,\mathrm{output}=''list''\right)$

$\left[-\mathrm{D\_y}z\+\mathrm{D\_z}y\,\mathrm{D\_z}\,-\mathrm{D\_x}z\+\mathrm{D\_z}x\,-\mathrm{D\_x}y\+\mathrm{D\_y}x\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$X≔\mathrm{evalDG}\left(R\left(x\,y\,z\right)\mathrm{D\_x}+S\left(x\,y\,z\right)\mathrm{D\_y}+T\left(x\,y\,z\right)\mathrm{D\_z}\right)$

$X:=R\left(x\,y\,z\right)\mathrm{D\_x}\+S\left(x\,y\,z\right)\mathrm{D\_y}\+T\left(x\,y\,z\right)\mathrm{D\_z}$

$U≔\left[R\left(x\,y\,z\right)\,S\left(x\,y\,z\right)\,T\left(x\,y\,z\right)\right]$

$U:=\left[R\left(x\,y\,z\right)\,S\left(x\,y\,z\right)\,T\left(x\,y\,z\right)\right]$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right]\,\mathrm{output}=''pde''\,\mathrm{unknowns}=U\,\mathrm{ansatz}=X\right)$

$\left[2\left(\frac{\partial }{\partial x}R\left(x\,y\,z\right)\right)\,\frac{\partial }{\partial x}S\left(x\,y\,z\right)\+\frac{\partial }{\partial y}R\left(x\,y\,z\right)\,\frac{\partial }{\partial x}T\left(x\,y\,z\right)\+\frac{\partial }{\partial z}R\left(x\,y\,z\right)\,\frac{\partial }{\partial x}S\left(x\,y\,z\right)\+\frac{\partial }{\partial y}R\left(x\,y\,z\right)\,2\left(\frac{\partial }{\partial y}S\left(x\,y\,z\right)\right)\,\frac{\partial }{\partial y}T\left(x\,y\,z\right)\+\frac{\partial }{\partial z}S\left(x\,y\,z\right)\,\frac{\partial }{\partial x}T\left(x\,y\,z\right)\+\frac{\partial }{\partial z}R\left(x\,y\,z\right)\,\frac{\partial }{\partial y}T\left(x\,y\,z\right)\+\frac{\partial }{\partial z}S\left(x\,y\,z\right)\,2\left(\frac{\partial }{\partial z}T\left(x\,y\,z\right)\right)\,0\right]\,\left[R\left(x\,y\,z\right)\,S\left(x\,y\,z\right)\,T\left(x\,y\,z\right)\right]$

$\mathrm{\Delta }≔\left[R\left(x\,y\,z\right)+S\left(x\,y\,z\right)+T\left(x\,y\,z\right)=0\right]$

$\mathrm{\Δ}:=\left[R\left(x\,y\,z\right)\+S\left(x\,y\,z\right)\+T\left(x\,y\,z\right)\=0\right]$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right]\,\mathrm{output}=''list''\,\mathrm{auxiliaryequations}=\mathrm{\Delta }\,\mathrm{unknowns}=U\,\mathrm{ansatz}=X\right)$

$\left[-\mathrm{D\_x}\+\mathrm{D\_z}\,-\left(y-z\right)\mathrm{D\_x}\+\left(x-z\right)\mathrm{D\_y}-\left(x-y\right)\mathrm{D\_z}\,-\mathrm{D\_x}\+\mathrm{D\_y}\right]$

$C≔\mathrm{Connection}\left(0\&mult\left(\left(\mathrm{D\_x}\&tensor\mathrm{dx}\right)\&tensor\mathrm{dx}\right)\right)$

$C:=0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}$

$\mathrm{\mu }≔\mathrm{evalDG}\left(\left(\mathrm{dx}\&w\mathrm{dy}\right)\&w\mathrm{dz}\right)$

$\mathrm{\μ}:=\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{\mu }\,C\right]\right)$

$-\left(\mathrm{\_C2}x\+\mathrm{\_C8}x-\mathrm{\_C9}z-y\mathrm{\_C11}-\mathrm{\_C10}\right)\mathrm{D\_x}\+\left(\mathrm{\_C5}x\+\mathrm{\_C6}z\+\mathrm{\_C8}y\+\mathrm{\_C7}\right)\mathrm{D\_y}\+\left(\mathrm{\_C1}x\+\mathrm{\_C2}z\+\mathrm{\_C4}y\+\mathrm{\_C3}\right)\mathrm{D\_z}$

$\mathrm{DGsetup}\left(\left[\mathrm{v1}\,\mathrm{v3}\,\mathrm{v4}\,\mathrm{w1}\,\mathrm{w3}\,\mathrm{w4}\,u\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{\omega }≔\mathrm{evalDG}\left(\left(\mathrm{dw3}\&w\mathrm{du}\right)\&w\mathrm{dv3}+\left(\mathrm{dv4}\&w\mathrm{du}\right)\&w\mathrm{dw4}+\left(\mathrm{dw1}\&w\mathrm{du}\right)\&w\mathrm{dv1}+2\left(\mathrm{dv1}\&w\mathrm{dv3}\right)\&w\mathrm{dw4}+2\left(\mathrm{dw1}\&w\mathrm{dw3}\right)\&w\mathrm{dv4}\right)$

$\mathrm{\ω}:=2\mathrm{dv1}\bigwedge \mathrm{dv3}\bigwedge \mathrm{dw4}\+\mathrm{dv1}\bigwedge \mathrm{dw1}\bigwedge \mathrm{du}\+\mathrm{dv3}\bigwedge \mathrm{dw3}\bigwedge \mathrm{du}\+2\mathrm{dv4}\bigwedge \mathrm{dw1}\bigwedge \mathrm{dw3}-\mathrm{dv4}\bigwedge \mathrm{dw4}\bigwedge \mathrm{du}$

$A≔\mathrm{Matrix}\left(7\,7\,\left(i\,j\right)\mapsto a\Vert i\Vert j\right)$

$X≔\mathrm{convert}\left(A\,\mathrm{DGvector}\right)$

$X:=\left(\mathrm{a11}\mathrm{v1}\+\mathrm{a12}\mathrm{v3}\+\mathrm{a13}\mathrm{v4}\+\mathrm{a14}\mathrm{w1}\+\mathrm{a15}\mathrm{w3}\+\mathrm{a16}\mathrm{w4}\+\mathrm{a17}u\right)\mathrm{D\_v1}\+\left(\mathrm{a21}\mathrm{v1}\+\mathrm{a22}\mathrm{v3}\+\mathrm{a23}\mathrm{v4}\+\mathrm{a24}\mathrm{w1}\+\mathrm{a25}\mathrm{w3}\+\mathrm{a26}\mathrm{w4}\+\mathrm{a27}u\right)\mathrm{D\_v3}\+\left(\mathrm{a31}\mathrm{v1}\+\mathrm{a32}\mathrm{v3}\+\mathrm{a33}\mathrm{v4}\+\mathrm{a34}\mathrm{w1}\+\mathrm{a35}\mathrm{w3}\+\mathrm{a36}\mathrm{w4}\+\mathrm{a37}u\right)\mathrm{D\_v4}\+\left(\mathrm{a41}\mathrm{v1}\+\mathrm{a42}\mathrm{v3}\+\mathrm{a43}\mathrm{v4}\+\mathrm{a44}\mathrm{w1}\+\mathrm{a45}\mathrm{w3}\+\mathrm{a46}\mathrm{w4}\+\mathrm{a47}u\right)\mathrm{D\_w1}\+\left(\mathrm{a51}\mathrm{v1}\+\mathrm{a52}\mathrm{v3}\+\mathrm{a53}\mathrm{v4}\+\mathrm{a54}\mathrm{w1}\+\mathrm{a55}\mathrm{w3}\+\mathrm{a56}\mathrm{w4}\+\mathrm{a57}u\right)\mathrm{D\_w3}\+\left(\mathrm{a61}\mathrm{v1}\+\mathrm{a62}\mathrm{v3}\+\mathrm{a63}\mathrm{v4}\+\mathrm{a64}\mathrm{w1}\+\mathrm{a65}\mathrm{w3}\+\mathrm{a66}\mathrm{w4}\+\mathrm{a67}u\right)\mathrm{D\_w4}\+\left(\mathrm{a71}\mathrm{v1}\+\mathrm{a72}\mathrm{v3}\+\mathrm{a73}\mathrm{v4}\+\mathrm{a74}\mathrm{w1}\+\mathrm{a75}\mathrm{w3}\+\mathrm{a76}\mathrm{w4}\+\mathrm{a77}u\right)\mathrm{D\_u}$

$\mathrm{vars}≔\mathrm{convert}\left(A\,\mathrm{set}\right)$

$\mathrm{vars}:=\left\{\mathrm{a11}\,\mathrm{a12}\,\mathrm{a13}\,\mathrm{a14}\,\mathrm{a15}\,\mathrm{a16}\,\mathrm{a17}\,\mathrm{a21}\,\mathrm{a22}\,\mathrm{a23}\,\mathrm{a24}\,\mathrm{a25}\,\mathrm{a26}\,\mathrm{a27}\,\mathrm{a31}\,\mathrm{a32}\,\mathrm{a33}\,\mathrm{a34}\,\mathrm{a35}\,\mathrm{a36}\,\mathrm{a37}\,\mathrm{a41}\,\mathrm{a42}\,\mathrm{a43}\,\mathrm{a44}\,\mathrm{a45}\,\mathrm{a46}\,\mathrm{a47}\,\mathrm{a51}\,\mathrm{a52}\,\mathrm{a53}\,\mathrm{a54}\,\mathrm{a55}\,\mathrm{a56}\,\mathrm{a57}\,\mathrm{a61}\,\mathrm{a62}\,\mathrm{a63}\,\mathrm{a64}\,\mathrm{a65}\,\mathrm{a66}\,\mathrm{a67}\,\mathrm{a71}\,\mathrm{a72}\,\mathrm{a73}\,\mathrm{a74}\,\mathrm{a75}\,\mathrm{a76}\,\mathrm{a77}\right\}$

$Y≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\mathrm{\omega }\,\mathrm{ansatz}=X\,\mathrm{unknowns}=\mathrm{vars}\right)$

$Y:=\left(\mathrm{\_C1}\mathrm{v1}\+\mathrm{\_C2}\mathrm{v3}\+\mathrm{\_C3}\mathrm{v4}\+\mathrm{\_C4}\mathrm{w3}\+\mathrm{\_C5}\mathrm{w4}\+\mathrm{\_C6}u\right)\mathrm{D\_v1}-\left(\mathrm{\_C4}\mathrm{w1}-\mathrm{\_C7}\mathrm{v1}-\mathrm{\_C8}\mathrm{v3}-\mathrm{\_C9}\mathrm{v4}-u\mathrm{\_C11}-\mathrm{w4}\mathrm{\_C10}\right)\mathrm{D\_v3}\+\left(\mathrm{\_C1}\mathrm{v4}\+\mathrm{\_C4}u-\mathrm{\_C5}\mathrm{w1}\+\mathrm{\_C6}\mathrm{v3}\+\mathrm{\_C8}\mathrm{v4}-\mathrm{v1}\mathrm{\_C11}-\mathrm{w3}\mathrm{\_C10}\right)\mathrm{D\_v4}-\left(\mathrm{\_C1}\mathrm{w1}\+\mathrm{\_C7}\mathrm{w3}-\mathrm{\_C9}u-\mathrm{v3}\mathrm{\_C12}-\mathrm{v4}\mathrm{\_C13}-\mathrm{w4}\mathrm{\_C11}\right)\mathrm{D\_w1}-\left(\mathrm{\_C2}\mathrm{w1}\+\mathrm{\_C3}u\+\mathrm{\_C6}\mathrm{w4}\+\mathrm{\_C8}\mathrm{w3}\+\mathrm{v1}\mathrm{\_C12}-\mathrm{v4}\mathrm{\_C14}\right)\mathrm{D\_w3}-\left(\mathrm{\_C1}\mathrm{w4}\+\mathrm{\_C3}\mathrm{w1}\+\mathrm{\_C8}\mathrm{w4}\+\mathrm{\_C9}\mathrm{w3}\+u\mathrm{\_C12}\+\mathrm{v1}\mathrm{\_C13}\+\mathrm{v3}\mathrm{\_C14}\right)\mathrm{D\_w4}-\left(2\mathrm{\_C3}\mathrm{v3}-2\mathrm{\_C4}\mathrm{w4}-2\mathrm{\_C6}\mathrm{w1}-2\mathrm{\_C9}\mathrm{v1}\+2\mathrm{v4}\mathrm{\_C12}-2\mathrm{w3}\mathrm{\_C11}\right)\mathrm{D\_u}$

$c≔\mathrm{Tools}:-\mathrm{DGinfo}\left(Y\,''NonJetIndets''\right)$

$c:=\left\{\mathrm{\_C1}\,\mathrm{\_C2}\,\mathrm{\_C3}\,\mathrm{\_C4}\,\mathrm{\_C5}\,\mathrm{\_C6}\,\mathrm{\_C7}\,\mathrm{\_C8}\,\mathrm{\_C9}\,\mathrm{\_C10}\,\mathrm{\_C11}\,\mathrm{\_C12}\,\mathrm{\_C13}\,\mathrm{\_C14}\right\}$

$\mathrm{Gamma}≔\left[\mathrm{seq}\left(\mathrm{Tools}:-\mathrm{DGmap}\left(1\,\mathrm{diff}\,Y\,v\right)\,v=c\right)\right]$

$\mathrm{\Γ}:=\left[\mathrm{v1}\mathrm{D\_v1}\+\mathrm{v4}\mathrm{D\_v4}-\mathrm{w1}\mathrm{D\_w1}-\mathrm{w4}\mathrm{D\_w4}\,\mathrm{v3}\mathrm{D\_v1}-\mathrm{w1}\mathrm{D\_w3}\,-2\mathrm{D\_u}\mathrm{v3}-u\mathrm{D\_w3}\+\mathrm{v4}\mathrm{D\_v1}-\mathrm{w1}\mathrm{D\_w4}\,2\mathrm{D\_u}\mathrm{w4}\+u\mathrm{D\_v4}-\mathrm{w1}\mathrm{D\_v3}\+\mathrm{w3}\mathrm{D\_v1}\,-\mathrm{w1}\mathrm{D\_v4}\+\mathrm{w4}\mathrm{D\_v1}\,2\mathrm{D\_u}\mathrm{w1}\+u\mathrm{D\_v1}\+\mathrm{v3}\mathrm{D\_v4}-\mathrm{w4}\mathrm{D\_w3}\,\mathrm{v1}\mathrm{D\_v3}-\mathrm{w3}\mathrm{D\_w1}\,\mathrm{v3}\mathrm{D\_v3}\+\mathrm{v4}\mathrm{D\_v4}-\mathrm{w3}\mathrm{D\_w3}-\mathrm{w4}\mathrm{D\_w4}\,2\mathrm{D\_u}\mathrm{v1}\+u\mathrm{D\_w1}\+\mathrm{v4}\mathrm{D\_v3}-\mathrm{w3}\mathrm{D\_w4}\,-\mathrm{w3}\mathrm{D\_v4}\+\mathrm{w4}\mathrm{D\_v3}\,2\mathrm{D\_u}\mathrm{w3}\+u\mathrm{D\_v3}-\mathrm{v1}\mathrm{D\_v4}\+\mathrm{w4}\mathrm{D\_w1}\,-2\mathrm{D\_u}\mathrm{v4}-u\mathrm{D\_w4}-\mathrm{v1}\mathrm{D\_w3}\+\mathrm{v3}\mathrm{D\_w1}\,-\mathrm{v1}\mathrm{D\_w4}\+\mathrm{v4}\mathrm{D\_w1}\,-\mathrm{v3}\mathrm{D\_w4}\+\mathrm{v4}\mathrm{D\_w3}\right]$

$\mathrm{nops}\left(\mathrm{Gamma}\right)$

$14$

$\mathrm{DGsetup}\left(\left[x\,y\,t\right]\,\left[u\right]\,J\,1\right)$

$\mathrm{frame\; name:\; J}$

$\mathrm{\lambda }≔\mathrm{evalDG}\left(\left({u\left[1\right]}^2+{u\left[2\right]}^2-{u\left[3\right]}^2\right)\left(\mathrm{Dx}\&w\mathrm{Dy}\right)\&w\mathrm{Dt}\right)$

$\mathrm{\λ}:=\left(u_1^2\+u_2^2-u_3^2\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge \mathrm{Dt}$

$\mathrm{Gamma}≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{\lambda }\right]\,\mathrm{output}=''list''\right)$

$\mathrm{\Γ}:=\left[-2\mathrm{D\_t}t-2\mathrm{D\_x}x-2\mathrm{D\_y}y\+{\mathrm{D\_u}}_{\[\]}{u}_{\[\]}\,{\mathrm{D\_u}}_{\[\]}\,\mathrm{D\_t}y\+\mathrm{D\_y}t\,\mathrm{D\_t}\,\mathrm{D\_t}x\+\mathrm{D\_x}t\,-\mathrm{D\_x}y\+\mathrm{D\_y}x\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$\mathrm{nops}\left(\mathrm{Gamma}\right)$

$8$

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

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$\mathrm{ConSym}≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\left[g\right]\right]\,\mathrm{output}=''list''\right)$

$\mathrm{ConSym}:=\left[\left(-\frac{1}{4}{y}^2-\frac{1}{4}{z}^2\+\frac{1}{4}{x}^2\right)\mathrm{D\_x}\+\frac{1}{2}xy\mathrm{D\_y}\+\frac{1}{2}xz\mathrm{D\_z}\,\frac{1}{2}xz\mathrm{D\_x}\+\frac{1}{2}yz\mathrm{D\_y}-\left(-\frac{1}{4}{z}^2\+\frac{1}{4}{x}^2\+\frac{1}{4}{y}^2\right)\mathrm{D\_z}\,\frac{1}{2}x\mathrm{D\_x}\+\frac{1}{2}y\mathrm{D\_y}\+\frac{1}{2}z\mathrm{D\_z}\,\frac{1}{2}xy\mathrm{D\_x}-\left(\frac{1}{4}{z}^2-\frac{1}{4}{y}^2\+\frac{1}{4}{x}^2\right)\mathrm{D\_y}\+\frac{1}{2}yz\mathrm{D\_z}\,-\mathrm{D\_y}z\+\mathrm{D\_z}y\,\mathrm{D\_z}\,-\mathrm{D\_x}z\+\mathrm{D\_z}x\,-\mathrm{D\_x}y\+\mathrm{D\_y}x\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

$\mathrm{nops}\left(\mathrm{ConSym}\right)$

$10$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\,\mathrm{x5}\right]\,Q\right)$

$\mathrm{frame\; name:\; Q}$

$\mathrm{\Delta }≔\mathrm{evalDG}\left(\left[\mathrm{D\_x1}+\mathrm{x3}\mathrm{D\_x2}+\mathrm{x4}\mathrm{D\_x3}+{\mathrm{x4}}^3\mathrm{D\_x5}\,\mathrm{D\_x4}\right]\right)\:$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{\Delta }\right]\,\mathrm{output}=''list''\right)$

$\left[-\mathrm{x2}\mathrm{D\_x2}-\mathrm{x3}\mathrm{D\_x3}-\mathrm{x4}\mathrm{D\_x4}-3\mathrm{x5}\mathrm{D\_x5}\,-\frac{1}{2}{\mathrm{x4}}^2\mathrm{D\_x1}-\left(-\frac{1}{6}\mathrm{x5}\+\frac{1}{2}\mathrm{x3}{\mathrm{x4}}^2\right)\mathrm{D\_x2}-\frac{1}{3}{\mathrm{x4}}^3\mathrm{D\_x3}-\frac{1}{5}{\mathrm{x4}}^5\mathrm{D\_x5}\,-\mathrm{x1}\mathrm{D\_x1}-2\mathrm{x2}\mathrm{D\_x2}-\mathrm{x3}\mathrm{D\_x3}-\mathrm{x5}\mathrm{D\_x5}\,\mathrm{D\_x5}\,\mathrm{x1}\mathrm{D\_x2}\+\mathrm{D\_x3}\,\mathrm{D\_x2}\,\mathrm{D\_x1}\right]$

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\exp \left(\mathrm{\alpha }x\right)\mathrm{dy}\&t\mathrm{dy}+\left(\mathrm{\beta }y+1\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dx}\mathrm{dx}\+{\ⅇ}^{\mathrm{\α}x}\mathrm{dy}\mathrm{dy}\+\left(y\mathrm{\β}\+1\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g\right]\,\mathrm{output}=''list''\,\mathrm{parameters}=\left\{\mathrm{\alpha }\,\mathrm{\beta }\right\}\,\mathrm{auxiliaryequations}=\left\{\mathrm{\alpha }\ne 0\right\}\right)$

$\left[\mathrm{D\_z}\,-\frac{2\mathrm{D\_x}}{\mathrm{\α}}\+y\mathrm{D\_y}\,\mathrm{D\_y}\,y\mathrm{\α}\mathrm{D\_x}-\left(\frac{1}{4}{y}^2{\mathrm{\α}}^2-{\ⅇ}^{-\mathrm{\α}x}\right)\mathrm{D\_y}\right]\,\left[\frac{4\mathrm{D\_x}}{\mathrm{\α}}-\frac{2\left(y\mathrm{\β}\+1\right)\mathrm{D\_y}}{\mathrm{\β}}\+z\mathrm{D\_z}\,\mathrm{D\_z}\right]\,\left[\left\{\mathrm{\α}\=\mathrm{\α}\,\mathrm{\β}\=0\right\}\,\left\{\mathrm{\α}\=\mathrm{\α}\,\mathrm{\β}\=\mathrm{\β}\right\}\right]$

$\mathrm{LD}≔\mathrm{Library}:-\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[4\,10\right]\,\mathrm{alg1}\right)$

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

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg1}$

$T≔\mathrm{evalDG}\left(\left(\mathrm{e4}\&t\mathrm{\θ1}\right)\&t\mathrm{e4}\right)$

$T:=\mathrm{e4}\mathrm{\θ1}\mathrm{e4}$

$\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[T\right]\right)$

$\mathrm{\_C1}\mathrm{e1}\+\mathrm{\_C2}\mathrm{e4}$