About the diagonalization of non-linear (in fields) actions

Suppose we have some interacting theory with the action:
$$
S = int d^{D} x left(partial_mu phi partial^mu phi + V(phi)right)
$$
Where $V(phi)$ is a potential (some polynomial of degree $>2$, $cosh$, whatever makes the equation of motion nonlinear). If one knew in principle a general solution of equation of motion, corresponding to the action above:
$$
phi(alpha)
$$
where $alpha$ is some set of parameters determining the solution, which for the case of quadratic action is the momentum $k$, would one obtain the quadratic action in the diagonal form:
$$
S = int d alpha phi(alpha) M(alpha) phi(alpha)
$$
Where $M(alpha)$ is some diagonal matrix, depending on these parameters $alpha$, or this action would be diagonalized in some other sense?

I am not aware of the interacting theories, where the general solution does exist (probably for my shame), for the case of $phi^4$ theory in $4D$ there is a Fubini instanton https://inspirehep.net/literature/108217, kinks in $2D$ field theory with double well potential, Sine-gordon model, instantons in the Yang-Mills theories, but these are rather special solutions.

Are there any examples of theories, where the generic solution is known?

Or the presence of such a potential would imply, that this theory is unsolvable in principle? And one can obtain a solution only in form of perturbation series?

I would be grateful for comments and suggestions!