In many cases of interest, $X$ is a coadjoint orbit of a Lie group $G$, and $H$ an element in the corresponding Lie-Poisson algebra of the Lie algebra of $G$.

These spaces describe in particular lots of exactly solvable problems - here $H$ is a sum of elements of the Lie algebra multiplied with Casimirs, plus a Casimir. Most nice exactly solvable problem can be cast in this form. See http://www.physicsoverflow.org/21556/coadjoint-orbits-in-physics?

For more on coadjoint orbits and their role in classical mechanics see

J.E. Marsden and T.S. Ratiu,

Introduction to mechanics and symmetry,

Springer, New York 1994.

http://libgen.org/search.php?req=Marsden+symmetry

(A short notice is also in http://en.wikipedia.org/wiki/Coadjoint_representation )