  We continue the study of the homogenization of coercive non-convex
Hamilton-Jacobi equations in random media identifying two general classes of
Hamiltonians with very distinct behavior. For the first class there is no
homogenization in a particular environment while for the second homogenization
takes place in environments with finite range dependence. Motivated by the
recent counter-example of Ziliotto, who constructed a coercive but non-convex
Hamilton-Jacobi equation with stationary ergodic random potential field for
which homogenization does not hold, we show that same happens for coercive
Hamiltonians which have a strict saddle-point, a very local property. We also
identify, based on the recent work of Armstrong and Cardaliaguet on the
homogenization of positively homogeneous random Hamiltonians in environments
with finite range dependence, a new general class Hamiltonians, namely
equations with uniformly strictly star-shaped sub-level sets, which homogenize.
