  In this letter I analyze a covering jet manifold scheme, its relation to the
invariant theory of the associated vector fields, and applications to the
Lax-Sato-type integrability of nonlinear dispersionless differential systems.
The related contact geometry linearization covering scheme is also discussed.
The devised techniques are demonstrated for such nonlinear Lax-Sato integrable
equations as Gibbons-Tsarev, ABC, Manakov-Santini and the differential Toda
singular manifold equations.
