Consider the PDE $DAD^Tu=0$ where $A$ is a constant matrix, $u:\mathbb{R}^n\rightarrow\mathbb{R}$ is twice differentiable and $D=[\frac{d}{dx_1},\frac{d}{dx_2}...]$.
I would like to find the group of all functions $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that if $u(x)$ is a solution to the PDE, then $u(g(x))$ is also a solution. For example, if $A$ is the identity matrix, then for any orthogonal matrix $B$ and vector $v$, $g(x)=Bx+v$ is such a $g$. Notice that this forms a group because if $u(g(x))$ is a solution iff $u(x)$ is a solution and $u(h(x))$ is a solution iff $u(x)$ is a solution, then $u(h(g(x)))$ is a solution when $u(x)$ is a solution.
