N=4 mechanics, WDVV equations and roots

N=4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation. While F cannot vanish and triggers translation non-invariance, U=0 yields a class of solutions (with vanishing central charge) which are encoded by the finite Coxeter systems. We extend previously known WDVV solutions in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A full classification strategy is proposed. The corresponding irreducible mechanics models admit a central-charge deformation (U\neq0) only for A_1 and in certain parts of the even dihedral systems I_2(2q). Thus, up to coordinate change and orthogonal composition, tunable couplings occur only in peculiar three-particle models, among which the one based on G_2+A_1 is the prime example. For vanishing central charge and any given F background, U may be constructed as a generalized hypergeometric function.