Compatible Poisson brackets of hydrodynamic type
  Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular:   (1) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor;   (2) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case;   (3) the connection of this problem with the class of surfaces in Euclidean space which possess nontrivial deformations preserving the Weingarten operator. 
E.V. Ferapontov