  We investigate algebraically coisotropic submanifolds $X$ in a holomorphic
symplectic projective manifold $M$. Motivated by our results in the
hypersurface case, we raise the following question: when $X$ is not uniruled,
is it true that up to a finite \'etale cover, the pair $(X,M)$ is a product
$(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subset
N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian
variety, and give some partial answer when the canonical bundle $K_X$ is
semi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$
(in fact this also holds without nefness assumption). We also remark that
Lagrangian submanifolds do not exist on a sufficiently general Abelian variety,
in contrast to the case when $M$ is irreducible hyperk\"ahler.
