Equiaffine Characterization Of Lagrangian Surfaces in $R^4$

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 $R^4$.