  In this paper we propose a geometric Hamilton--Jacobi theory on a
Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi theory
is that if a Hamiltonian vector field $X_H$ can be projected into a
configuration manifold by means of a one-form $dW$, then the integral curves of
the projected vector field $X_H^{dW}$ can be transformed into integral curves
of the vector field $X_H$ provided that $W$ is a solution of the
Hamilton--Jacobi equation. This procedure allows us to reduce the dynamics to a
lower dimensional manifold in which we integrate the motion. On the other hand,
the interest of a Nambu--Jacobi structure resides in its role in the
description of dynamics in terms of several Hamiltonian functions. It appears
in fluid dynamics, for instance. Here, we derive an explicit expression for a
geometric Hamilton--Jacobi equation on a Nambu--Jacobi manifold and apply it to
the third-order Riccati differential equation as an example.
