Bihamiltonian Systems and Lax Representation

Between the methods for obtaining first integrals of hamiltonian systems there are two that have acquired more and more relevance in the study of integrable hamiltonian systems, they are the following: (1). The representation of the system as a Lax Pair $$\dot L = \left[ {L,\,A} \right],$$ , where L, A belong to a Lie algebra. (2). The bihamiltonian method, which consist in the existence of an alternative Poisson bracket, such that the linear combinations of this bracket with the original one are a Poisson bracket too and the hamiltonian equation of the second bracket, with respect to some hamiltonian, is the same that the original one.