Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

The structure and properties of families of critical points for classes of functions W(z, z̄) obeying the elliptic Euler-Poisson-Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrodinger hierarchies, the inverse hierarchy and equations associated with the real-analytic Eisenstein series E(β, β̄; 1/2) among them. The specific bi-Hamiltonian structure of these equations is also discussed. © 2013 IOP Publishing Ltd Printed in the UK and the USA.