Lax representation and Poisson geometry of the Kowalevski top

We discuss the Poisson structure underlying the two-field Kowalevski gyrost at and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the s tructure of bi-Hamiltonian completely integrable systems. We study the Casi mir functions of such pencils, and show how it is possible to frame the Kow alevski systems within the so-called Gef'fand-Zakharevich bi-Hamiltonian se tting for integrable systems.