Differential-geometric structures defining Lie-Backlund transformations

The contact transformations of differential-geometric structures defining lie-bäcklund transformations are considered as differential-geometric structures by specifying the field of a geometric object. The study also highlighted the differential-geometric structure of second-order contact transformations over a two-dimensional basis. The diffeomorphism between the 2-jet manifolds J2E and J2E are specified by either explicit equations establishing relations between the local coordinates of the manifolds J2E and J1E or Pfaffian equations, which establish relations between the principal differential forms on these manifolds. The fundamental object of a 2-diffeomorphism contains the linear homogeneous subobject with components, where a second-order partial differential equation can be treated as an equation relating the different variables.