  The Bures metric and the associated Bures-Hall measure is arguably the best
choice for studying the spectrum of the quantum mechanical density matrix with
no apriori knowledge of the system. We investigate the probability of a gap in
the spectrum of this model, either at the bottom $ [0,s) $ or at the top $
(s,1] $, utilising the connection of this Pfaffian point-process with the
allied problem in the determinantal point-process of the two-dimensional
Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two
variables $s,t$. To this end we develop new general results about Cauchy
bi-orthogonal polynomial system for a more general class of weights than the
Laguerre densities: in particular a new Christoffel-Darboux formula,
reproducing kernels and differential equations for the polynomials and their
associated functions. This system is most simply expressed as rank-3 matrix
variables and possesses an associated cubic bilinear form. Furthermore under
specialisation to truncated Laguerre type densities for the weight, of direct
relevance to the Cauchy-Laguerre system, we construct a closed system of
constrained, nonlinear differential equations in two deformation variables
$s,t$, and observe that the recurrence, spectral and deformation derivative
structures form a compatible and integrable triplet of Lax equations.
