Strong Embeddability of Time-Invariant Nonlinear Differential Systems in Linear Differential Systems

We consider a pair (Σ, w ) consisting of a time-invariant nonlinear differential system Σ and a real piecewise smooth function w defined on the state space of the differential system. The notion of strong embeddability of the pair (Σ, w ) in a linear differential system is introduced. A criterion for the finite-dimensional-strong embeddability of the pair is obtained, linear embedding systems are described, and the explicit representation of first integrals is given for systems in finite-dimensional-strongly embeddable pairs. The class of first integrals determining the general solution of a finite-dimensional-strongly embeddable pair is singled out.