identification of new solvable dynamical systems of nonlinear odes

The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of algebraically solvable dynamical systems. recently this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; and subsequently significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. here we report an approach suitable to deal with the most general case, i. e. that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. we then focus on the special case of a polynomial featuring only 2 different zeros (of arbitrary multiplicity) and, by using a recently introduced additional twist of this approach, we thereby identify new classes of algebraically solvable dynamical systems described by 2 first-order odes in 2 dependent variables with polynomial right-hand sides. this is a joint work with prof. francesco calogero.