Integrable field theories from Poisson algebras

New integrable 1 + 1 dimensional classical field theories are found that include infinite dimensional analogues of N-particle Toda-and Calogero-Moser systems, as well as non-relativistic theories with an interaction that is polynomial in the first (spatial) derivative of the field. The existence, as well as the involutivity, of an infinite set of independent conserved quantities follows most easily from a 2 + 1 dimensional Lax-pair which uses as its underlying infinite dimensional Lie algebra a Poisson algebra of functions in two variables