On the space of $2d$ integrable models

We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We classify the commuting subalgebras of the $2d$ free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.