It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley
(BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess
solutions whose level surfaces are central quadrics in the space of independent
variables (the so-called central quadric ansatz). It was demonstrated that
generic solutions of this type are described by Painleve equations PIII and
PII, respectively. The aim of our paper is threefold:
-- Based on the method of hydrodynamic reductions, we classify integrable
models possessing the central quadric ansatz. This leads to the five canonical
forms (including BF and dKP).
-- Applying the central quadric ansatz to each of the five canonical forms,
we obtain all Painleve equations PI - PVI, with PVI corresponding to the
generic case of our classification.
-- We argue that solutions coming from the central quadric ansatz constitute
a subclass of two-phase solutions provided by the method of hydrodynamic
reductions.Comment: 12 page
