Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?
An integrable system is, at least, a map from X to R^n whose coordinate functions Poisson commute. The moment map of a Hamiltonian torus action will have this property, but there are other examples. For instance, the flag variety GL(n,C)/B has a famous integrable system and a famous toric degeneration, both of which are related to the same polytope--a Gelfand-Tsetlin polytope. (Famous but I don't know the original references for these constructions.)
Given a toric degeneration Y --> C, you can try to construct an integrable system on a general fiber Y1 by flowing along a gradient vector field from Y1 to Y0 (the special fiber, a toric variety) and projecting to R^n via the moment map of the torus action on Y0. I heard that this doesn't work on the nose, but that it does work well enough that you can at suitable points identify the fibers of e.g. the Gelfand-Tsetlin integrable system with the Milnor fibers of the Gelfand-Tsetlin toric degeneration. Possibly starting with an integrable system and trying to construct a toric degeneration is easier and more algebraic.
P.S. Some references after all: Guillemin and Sternberg, "The Gelfand-Cetlin system and quantization of the complex flag manifolds," and Gonciulea and Lakshmibai, "Degenerations of Flag and Schubert varieties to toric varieties."