On integrals, Hamiltonian and metriplectic formulations of polynomial systems in 3D

The first integrals of the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh–Rose model, and the Oregonator model are derived using the method of Darboux polynomials. It is shown that, the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh–Rose model can be written in a bi-Hamiltonian/Nambu metriplectic form.