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.