Let $L$ be any differential operator (not necessarily linear).

Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:

Given a boundary value problem on some domain $\Omega$. If $L$, the initial condition, and the boundary are of a certain form, then separation of variables will yield the solution to the boundary value problem.

Do such general statements exist, and if they do, where can I find them? I am interested in statements about strong (classical) solutions and weak (distributional) solutions.

I am aware that questions similar to this one are already posted on this site, however, all of them consider specific differential operators and then they focus on separability of the domain. I am interested in statements about general differential operators.