Rotationally invariant ensembles of integrable matrices

We construct ensembles of \textit{random integrable matrices} with any prescribed number of nontrivial integrals and formulate \textit{integrable matrix theory} (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-$M$ family of integrable matrices consists of exactly $N-M$ independent commuting $N\times N$ matrices linear in a real parameter. We first develop a rotationally invariant parameterization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice-versa. Higher types similarly involve a random vector and two matrices. The basis-independent formulation allows us to derive the joint probability density for integrable matrices, in a manner similar to the construction of Gaussian ensembles in the RMT.