In this article, we propose a dynamic model of the three-dimensional eel swim. This model is analytical and suited to the on-line control of eel-like robots. The proposed solution is based on the Large Amplitude Elongated Body Theory of Lighthill and a working frame recently proposed in [1] for the dynamic modeling of hyper-redundant robots. This working frame was named "macro-continuous" since at this macroscopic scale, the robot (or the animal) is considered as a Cosserat beam internally (and continuously) actuated. This article proposes new results in two directions. Firstly, it achieves an extension of the Lighthill theory to the case of a self propelled body swimming in three dimensions, while including a model of the internal control torque. Secondly, this generalization of the Lighthill model is achieved due to a new set of equations which is also derived in this article. These equations generalize the Poincaré equations of a Cosserat beam to the case of an open system containing a fluid stratified around the slender beam.