  A new solvable many-body problem is identified. It is characterized by
nonlinear Newtonian equations of motion ("acceleration equal force") featuring
one-body and two-body velocity-dependent forces "of goldfish type" which
determine the motion of an arbitrary number $N$ of unit-mass point-particles in
a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ of the $N$
coordinates of these moving particles are given by the $N$ eigenvalues of a
time-dependent $N\times N$ matrix $U(t)$ explicitly known in terms of the 2N
initial data $z_{n}(0)$ and $\dot{z}_{n}(0)$. This model comes in two different
variants, one featuring 3 arbitrary coupling constants, the other only 2; for
special values of these parameters all solutions are completely periodic with
the same period independent of the initial data ("isochrony"); for other
special values of these parameters this property holds up to corrections
vanishing exponentially as $t\rightarrow \infty$ ("asymptotic isochrony").
Other isochronous variants of these models are also reported. Alternative
formulations, obtained by changing the dependent variables from the $N$ zeros
of a monic polynomial of degree $N$ to its $N$ coefficients, are also
exhibited. Some mathematical findings implied by some of these results - such
as Diophantine properties of the zeros of certain polynomials - are outlined,
but their analysis is postponed to a separate paper.
