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In this work one proves that, around each point of a dense open set (regularpoints), a real analytic or holomorphic bihamiltonian structure decomposes intoa product of a Kronecker bihamiltonian structure and a symplectic one if anecessary condition on the characteristic polynomial of the symplectic factorholds. Moreover we give an example of bihamiltonian structure for showing thatthis result does not extend to the $C^\infty$-category.  Thus a classical problem on the geometric theory of bihamiltonian structuresis solved at almost every point.

The Local Product Theorem for bihamiltonian structures