Schrodinger equations on foliated spaces

Schroedinger type equations have been constantly explored in mathematics. I would like to know if is it possible to make sense of physical interpretation in the following setting:

One has a closed manifold $M$ (I mean, it is compact and has no boundary (such as a sphere)) with a group action by a Lie group $G$. Then, state a Schroedinger type equation on this manifold. On could possible ask to: are there $G$-invariant solutions for this equation?

More generally, assume that $M$ is foliated by sub manifolds and analogously pose a Schroedinger equation on it. One could ask: are there solutions that are constant along the leaves?