Generalized Lax pairs for the computation of semi-invariants

A Lax pair is a classical tool for the computation of first integrals of continuous-time nonlinear systems. Semi-invariants extend the concept of first integral and generalize the concept of the pair (eigenvalue, left eigenvector) of a linear mapping to the nonlinear framework, whence play the role of basic bricks for the computation of Lyapunov functions in closed-form. In this paper, it is shown how Lax pairs can be generalized to allow semi-invariants to be computed in an algebraic way. The geometric nature of this generalization allows a parallel treatment of both continuous-time and discrete-time systems.