  A new class of solvable $N$-body problems is identified. They describe $N$
unit-mass point particles whose time-evolution, generally taking place in the
complex plane, is characterized by Newtonian equations of motion "of goldfish
type" (acceleration equal force, with specific velocity-dependent one-body and
two-body forces) featuring several arbitrary coupling constants. The
corresponding initial-value problems are solved by finding the eigenvalues of a
time-dependent $N\times N$ matrix $U(t)$ explicitly defined in terms of the
initial positions and velocities of the $N$ particles. Some of these models are
asymptotically isochronous, i.e. in the remote future they become completely
periodic with a period $T$ independent of the initial data (up to exponentially
vanishing corrections). Alternative formulations of these models, obtained by
changing the dependent variables from the $N$ zeros of a monic polynomial of
degree $N$ to its $N$ coefficients, are also exhibited.
