  We construct ensembles of random integrable matrices with any prescribed
number of nontrivial integrals and formulate 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-by-N matrices linear in a real parameter. We first develop a
rotationally invariant parametrization 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.
