In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the unionin of disjoint intervals, first introduced by N.I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a proevious paper, that involves the construciton of a certain meromorphic function on a hyperelliptic Riemann surface. The method de scribed here is based on the general Riemann-Hilert scheme of the theory of integrable systems and will enable us to derive, in a versy strightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantaties involing the corresponding recurrence coffcients. Both of these equations wre obtained earlier by A. Magnus. In our aproach, however, we are able to go beyond Magnus's results by actualy solving the equatins in terms of the Riemann Θ-funcitons. We also show that the related Hankel determinant can be nterpreted as the relevantτ-function.