The geometry of certain systems of partial differential equations

Geometry of some classes of the partial differential equations systems of the first, second and third orders is studied relatively to given transformations of the independent variables $u^\alpha$, unknown functions $x^i$ and their derivatives up to the corresponding order. Under the geometry of the system is understood the geometry of the corresponding space with given fundamental geometrical object, having as components the coefficients of the system. The intrinsic objects of linear differential geometric and affine connections of these spaces are defined.