  We prove a generalized version of the Morse index theorem for geodesics
endowed with a non positive definite metric tensor (semi-Riemannian manifolds).
We apply the result to obtain lower estimates on the number of geodesics
joining two fixed non conjugate points in certain classes of manifolds. More
specifically, we consider semi-Riemannian manifolds $(M,\mathfrak g)$ admitting
a smooth distribution spanned by commuting Killing vector fields and containing
a maximal negative distribution for $\mathfrak$. In particular we obtain Morse
relations for stationary semi-Riemannian manifolds and for the {\em
G\"odel-type} manifolds.
