  Let $M$ be a non-compact connected manifold with a cocompact and properly
discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf
theorem for a bounded vector field on $M$ satisfying a mild condition on zeros.
As an application, we show that such a vector field must have infinitely many
zeros whenever $G$ is amenable and the Euler characteristic of $M/G$ is
non-zero.
