Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits $\O=O_1\times...\times O_N$ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on $\O$. As a result we get the so called classical Gaudin system on $\O$. The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.