CHAMP DE YANG-MILLS AVEC GROUPE DE JAUGE SU(2) (YANG-MILLS FIELD WITH SU(2) AS GAUGE GROUP)

In this paper we consider a hamiltonian system which, in a special case and under the gauge group SU(2), can be considered as some reduction of the Yang-Mills field equations. We study this system from a different angle. We show that this system is completely integrable and we realize explicitly, using the Lax spectral curve technique that the flows generated by the constants of the motion are straight lines on the Jacobi variety of a genus 2 hyperelliptic curve. We show that at some special values of the parameters, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation. We show that the complex affine variety defined by putting the invariants of the system equal to generic constants completes into an abelian surface and the system is algebraic completely integrable. We give a direct proof that the abelian surface obtained in this paper is dual to Prym variety of an hyperelliptic curve of genus 3.