Contactization of First Order PDE Systems

The article is devoted to applications of contact geometry to systems of first-order nonlinear partial differential equations (PDE). The traditional geometric approach involves consideration of such systems as submanifolds in the 1-jet space. Unlike scalar differential equations, contact transformations of systems are prolongations of point transformations. The main idea of this article is to expand the class of admissible transformations of systems of differential equations. To this end, we extend the even-dimensional space of dependent and independent variables to an odd-dimensional one and introduce a contact structure in this extended space. We call this procedure contactization of systems. Contact structures are determined by the conservation laws of systems and differ from the contact structure generated by the Cartan form on the space of 1-jets. This approach allows us to significantly expand the class of admissible transformations of systems and consider issues of integrability.