  Many finite dimensional integrable systems qre expressed with the help of the
Lax equation which highlights a spectral parameter and therefore a spectral
curve. These spectral curves are the starting point of an algebro-geometric
investigation. The algebraic aspect of this investigation will be expressed
using the Jacobians of smooth spectral curves and generalized jacobians of
singular spectral curves and in the other hand the geometrical aspect will be
attributed to the vector fields which are defined by the Lax equation. In this
article will be dedicated only to singular hyperelliptic spectral curves that
are associated to Mumford systems. We will study the complementarity of these
two mathematical approaches to describe the same object and revealing their
characteristics.
