  For non-degenerate surfaces in $R^4$, a distinguished transversal bundle
called affine normal plane bundle was proposed in [Nomizu-Vrancken]. Lagrangian
surfaces have remarkable properties with respect to this normal bundle, like
for example, the normal bundle being Lagrangian. In this paper we characterize
those surfaces which are Lagrangian with respect to some parallel symplectic
form in $R^4$.
