On the central quadric ansatz: integrable models and Painleve reductions
  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. 
E.V. Ferapontov, B. Huard, A. Zhang
nlin.SI math-ph math.DG math.MP