A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure

The Hamilton-Jacobi theory is a formulation of Classical Mechanics equivalent to other formulations as Newton's equations, Lagrangian or Hamiltonian Mechanics. It is particulary useful for the identification of conserved quantities of a mechanical system. The primordial observation of a geometric Hamilton-Jacobi equation is that if a Hamiltonian vector field $X_{H}$ can be projected into the configuration manifold by means of a 1-form $dW$, then the integral curves of the projected vector field $X_{H}^{dW}$can be transformed into integral curves of $X_{H}$ provided that $W$ is a solution of the Hamilton-Jacobi equation. This interpretation has been applied to multiple settings: in nonhonolomic, singular Lagrangian Mechanics and classical field theories. Our aim is to apply the geometric Hamilton-Jacobi theory to systems endowed with a Nambu-Poisson structure. The Nambu-Poisson structure has shown its interest in the study physical systems described by several Hamiltonian functions. In this way, we will apply our theory to two interesting examples in the Physics literature: the third-order Kummer-Schwarz equations and a system of $n$ copies of a first-order differential Riccati equation. From these examples, we retrieve the original Nambu bracket in three dimensions and a generalization of the Nambu bracket to $n$ dimensions, respectively.