  We develop an approach to affine symplectic invariant geometry of Lagrangian
surfaces by the method of moving frames. The fundamental invariants of elliptic
Lagrangian immersions in affine symplectic four-space are derived together with
their integrability equations. The invariant setup is applied to discuss the
question of symplectic applicability for elliptic Lagrangian immersions.
Explicit examples are considered.
