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The theory of triples of Poisson brackets and related integrable systems,based on a classical R-matrix R in End_F(g), where g is a finite dimensionalassociative algebra over a field F viewed as a Lie algebra, was developed byOevel-Ragnisco and Li-Parmentier [OR89,LP89].  In the present paper we develop an "affine" analogue of this theory byintroducing the notion of a continuous Poisson vertex algebra and constructingtriples of Poisson lambda-brackets. We introduce the corresponding Adler typeidentities and apply them to integrability of hierarchies of Hamiltonian PDEs.

Adler-Oevel-Ragnisco type operators and Poisson vertex algebras