  We define hypersymplectic structures on Lie algebroids recovering, as
particular cases, all the classical results and examples of hypersymplectic
structures on manifolds. We prove a 1-1 correspondence theorem between
hypersymplectic structures and (pseudo-)hyperk\"{a}hler structures. We show
that the hypersymplectic framework is very rich in already known compatible
pairs of tensors such as Poisson-Nijenhuis, $\Omega N$ and $P \Omega$
structures.
