  An attempt is made to describe random matrix ensembles with unitary
invariance of measure (UE) in a unified way, using a combination of Tracy-Widom
(TW) and Adler-Shiota-Van Moerbeke (ASvM) approaches to derivation of partial
differential equations (PDE) for spectral gap probabilities. First, general
3-term recurrence relations for UE restricted to subsets of real line, or, in
other words, for functions in the resolvent kernel, are obtained. Using them,
simple universal relations between all TW dependent variables and
one-dimensional Toda lattice $\tau$-functions are found. A universal system of
PDE for UE is derived from previous relations, which leads also to a {\it
single independent PDE} for spectral gap probability of various UE.
  Thus, orthogonal function bases and Toda lattice are seen at the core of
correspondence of different approaches. Moreover, Toda-AKNS system provides a
common structure of PDE for unitary ensembles. Interestingly, this structure
can be seen in two very different forms: one arises from orthogonal
functions-Toda lattice considerations, while the other comes from Schlesinger
equations for isomonodromic deformations and their relation with TW equations.
The simple example of Gaussian matrices most neatly exposes this structure.
