Description of compatible differential-geometric Poisson brackets of the   first order
  Compatible local differential-geometric Poisson brackets of the first order (Dubrovin-Novikov type) are classified by solutions of modified Sinh-Gordon equation. It was proved by E.V. Ferapontov. In this paper integrable system describing compatible non-local differential-geometric Poisson brackets of the first order (Ferapontov type) is presented in case when one metric is flat and another one has co-dimension 1. This is a reduction of the Cherednik model (chiral fields). 
Maxim Pavlov