The Painleve property for quasihomogenous systems and a many-body problem in the plane

We investigate a many-body problem in the plane introduced by Calogero and intensively studied by Calogero, Francoise and Sommacal. An ad hoc complexification transforms the many-body problem to a system of second order autonomous complex equations depending on some complex constants that describe the two-body interactions. We investigate the sets of two-body interaction constants that make the complexified equation have the Painleve Property, this is, its solutions are given by single-valued meromorphic functions. In this case the original system has only periodic isochronous solutions. We exhibit a family of settings where the system displays this property and show that it is not present in the three- and four-body problems that do not fall within our class. For this, we introduce a necessary condition for the presence of the Painleve Property in some quasihomogenous systems.