  Recent years have seen an upsurge of interest in dynamical realizations of
the superconformal group SU(1,1|2) in mechanics. Remarking that SU(1,1|2) is a
particular member of a chain of supergroups SU(1,1|n) parametrized by an
integer n, here we begin a systematic study of SU(1,1|n) multi-particle
mechanics. A representation of the superconformal algebra su(1,1|n) is
constructed on the phase space spanned by m copies of the (1,2n,2n-1)
supermultiplet. We show that the dynamics is governed by two prepotentials V
and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a
consequence of a more general fourth-order equation. All solutions to the
latter in terms of root systems reveal decoupled models only. An extension of
the dynamical content of the (1,2n,2n-1) supermultiplet by angular variables in
a way similar to the SU(1,1|2) case is problematic.
