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We reconstruct a rational Lax matrix of size R+1 from its spectral curve (thedesingularization 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 ofbases of dual differentials of weights nu and 1-nu respectively. All objectsare described in the most explicit terms using Theta functions. Via a sequenceof ``elementary twists'', we construct sequences of Lax matrices sharing thesame spectral curve and polar structure and related by conjugations by rationalmatrices. Particular choices of elementary twists lead to construction ofsequences of Lax matrices related to finite--band recurrence relations (i.e.difference operators) sharing the same shape. Recurrences of this kind aresatisfied by several types of orthogonal and biorthogonal polynomials. Therelevance of formulae obtained to the study of the large degree asymptotics forthese polynomials is indicated.

Effective inverse spectral problem for rational Lax matrices and applications