  We consider the spectral problem of the Lax pair associated to periodic
integrable partial differential equations. We assume this spectral problem to
be a polynomial of degree $d$ in the spectral parameter $\lambda$. From this
assumption, we find the conservation laws as well as the hyperelliptic curve
required to solve the periodic inverse problem. A recursion formula is
developed, as well as $d$ additional conditions which give additional
information to integrate the equations under consideration. We also include two
examples to show how the techniques developed work. For the Korteweg-deVries
(KdV) equation, the degree of the multiplicative equation is $d=1$. Hence, we
only have one condition and one recursion formula. The condition gives in each
degree of the recursion the conserved densities for KdV equation, recovering
the Lax hierarchy. For the Nonlinear Schr\"odinger (NLS) equation, the degree
of the multiplicative operator is $d=2$. Hence, we have a couple of conditions
that we use to deduce conserved density constants and the Lax hierarchy for
such equation. Additionally, we explicitely write down the hyper-elliptic curve
associted to the NLS equation. Our approach can be use for other completely
integrable differential equations, as long as we have a polynomial
multiplicative operator associated to them.
