Hyperelliptic reduction of the Benney moment equations

We consider N-parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) to correspond to N-parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Probl. 16 605) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function is given by the inversion of a second kind Abelian integral on the Θ-divisor. This is found explicitly following a method given by Enolskii et al (2003 J. Nonlinear Sci. 13 157)