  We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the
desingularization of the characteristic polynomial) and some additional data.
Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu))
we provide a residue-formula for the entries of the Lax matrix in terms of
bases of dual differentials of weights nu and 1-nu respectively. All objects
are described in the most explicit terms using Theta functions. Via a sequence
of ``elementary twists'', we construct sequences of Lax matrices sharing the
same spectral curve and polar structure and related by conjugations by rational
matrices. Particular choices of elementary twists lead to construction of
sequences of Lax matrices related to finite--band recurrence relations (i.e.
difference operators) sharing the same shape. Recurrences of this kind are
satisfied by several types of orthogonal and biorthogonal polynomials. The
relevance of formulae obtained to the study of the large degree asymptotics for
these polynomials is indicated.
