  A time-dependent monic polynomial in the z variable with N distinct roots
such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the
k-th derivatives of the N roots are expressed in terms of the derivatives of
order j<= k of the first N coefficients of the polynomial and of the
derivatives of order j<= k-1 of the roots themselves. These relations are
utilized to construct new classes of algebraically solvable first order systems
of ODEs as well as N-body problems. Multiple examples of solvable isochronous
(all solutions are periodic with the same period) 2- and 3-body problems are
provided.
