  The aim of this paper is to study the natural action of the real symplectic
group, $\operatorname{Sp}(4, \mathbb{R})$, on the algebraic set of
$4$-dimensional Lie algebras admitting symplectic structures and to give a
complete classification of orbit closures.
  We present some applications of such classification to the study of the Ricci
curvature of left-invariant almost K\"{a}hler structures on four dimensional
Lie groups.
