Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation that has a different covariance group?
In other words: consider some solution to the linear Klein-Gordon equation, and model that solution by a mapping $\psi : M \rightarrow \mathbb{R}$, where $M = (\mathbb{R}^4, \eta)$ is Minkowski spacetime. Now consider a coordinate transformation---but not necessarily a Lorentz transformation. Instead, consider any coordinate transformation modeled by an autohomeomorphism $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$, where $\mathbb{R}^4$ has the standard induced topology. This gives $\psi'(x', t') : \mathbb{R}^4 \rightarrow \mathbb{R}$. The question is this: if one has free choice of metric on the new coordinates established by $\phi$, is there such a $\psi'$ that solves another equation with some different covariance group?
Further, if this is possible for simple, plane-wave solutions, then what are the conditions under which it is no longer possible? What about linear combinations of plane-wave solutions? What about the case of multiple solutions, the values of one being coincident with the values of the other(s) in $M$?
(Note: empirical adequacy is irrelevant; this is just a question about the conditions under which the values of a scalar-field solution to a partially-differential equation with some particular covariance group can also act as the solution to another equation with some different covariance group, while preserving their topological (or perhaps even differential) structure.)