Boundary Lax pairs from non-ultra local Poisson algebras

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We consider non-ultra local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or "boundary" extensions. They are parametrized by a "boundary" scalar matrix and depend in addition on the choice of an anti-automorphism. The new algebras are the classical-linear counterparts of known quadratic quantum boundary algebras. For any choice of parameters the non-ultra local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary PCM model is examined as a physical example.