Any model integrable but not separable?

Question

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?

Answer 1

The Toda-Lattice is integrable but not seperable https://en.wikipedia.org/wiki/Toda_lattice#Integrability

Im still learning all of this so I‘am unable to tell you why, but that‘s the example my professor gave.