Connections are exposed between integrable equations for spectral gap
probabilities of unitary invariant ensembles of random matrices (UE)
derived by different --- Tracy-Widom (TW) and Adler-Shiota-van Moerbeke
(ASvM) --- methods. Simple universal relations are obtained between these
probabilities and their ratios on one side, and variables of the approach
using resolvent kernels of Fredholm operators on the other side. A unified
description of UE is developed in terms of universal, i.e. independent of
the specific probability measure, PDEs for gap probabilities, using the
correspondence of TW and ASvM variables. These considerations are based on
the three-term recurrence for orthogonal polynomials (OP) and
one-dimensional Toda lattice (or Toda-AKNS) integrable hierarchy whose
flows are the continuous transformations between different OP bases.
Similar connections exist for coupled UE.
The gap probabilities for one-matrix Gaussian UE (GUE) or joint gap
probabilities for coupled GUE satisfy various PDEs whose number grows
with the number of spectral endpoints. With the above connections serving
as a guide, minimal complete sets of independent lowest order PDEs for
the GUE and for the largest eigenvalues of two-matrix coupled GUE are
found.