Symplectic and contact Lie algebras with an application to Monge-Ampere equations

In this paper we consider symplectic and contact Lie algebras. We define contactization and symplectization procedures and describe its main properties. We also give classification of such algebras in dimensions 3 and 4. The classification in dimension~4 is closely connected with normal forms of nondegenerate elliptic equations of the second order on two-dimensional surfaces with transitive symmetry group in first jets. We point out this connection and discuss normal forms.