  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.
