Sala Conferenze (Puteano, Centro De Giorgi).
We present a new notion of homogeneity for rational potentials in the plane, we called rotation homogeneity. These potentials can be seen as singular limits of rational potentials, and in particular, a necessary condition for integrability is that the dominant term and lower order term should be integrable. This new notion of homogeneity can combine with the classical one, producing a bihomogeneous case, which is a 2-integer parameter family. After reduction, the system becomes a Lotka Voltera quadratic planar vector field, and with a result of M Ollagnier, allows us to make complete classification of integrable cases. The reduction being constructive, we then explicitly integrate an exceptional case with a polynomial first integral of degree 4 in momenta using Abelian functions.