A solvable many-body problem, its equilibria, and a second-order ordinary differential equation whose general solution is polynomial.

Some properties of a solvable N-body problem featuring several free parameters ('coupling constants') are investigated. Restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data (isochrony). The restrictions on its parameters which guarantee the existence of equilibria are also identified. In this connection a remarkable second-order ODE-generally not of hypergeometric type, hence not reducible to those characterizing the classical polynomials-is studied: if its parameters satisfy a Diophantine condition, its general solution is a polynomial of degree N, the N zeros of which identify the equilibria of the N-body system. [ABSTRACT FROM AUTHOR]