  This paper addresses the classification problem of integrable deformations of
solutions of "degenerate" Riemann-Hilbert-Birkhoff (RHB) problems. These
consist of those RHB problems whose initial datum has diagonal pole part with
coalescing eigenvalues. On the one hand, according to theorems of B.Malgrange,
M.Jimbo, T.Miwa, and K.Ueno, in the non-degenerate case, there exists a
universal integrable deformation inducing (via a unique map) all other
deformations. On the other hand, in the degenerate case, C.Sabbah proved
(arXiv:1711.08514v4), under sharp conditions, the existence of an integrable
deformation of solutions, sharing many properties of the one constructed by
Malgrange-Jimbo-Miwa-Ueno. Albeit the integrable deformation constructed by
Sabbah is not, stricto sensu, universal, we prove that it satisfies a
$relative$ universal property. We show the existence and uniqueness of a
maximal class of integrable deformations all induced (via a unique map) by
Sabbah's integrable deformation. Furthermore, we show that such a class is
large enough to include all generic integrable deformations whose pole and
deformation parts are locally holomorphically diagonalizable. In itinere, we
also obtain a characterization of holomorphic matrix-valued maps which are
locally holomorphically Jordanizable. This extends, to the case of several
complex variables, already known results independently obtained by
Ph.G.A.Thijsse and W.Wasow.
