Darboux polynomial matrices: the classical Massive Thirring Model as study case

One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to obtain such matrices in polynomial form. Our method is illustrated by applying it to the classical Massive Thirring Model (MTM), and by combining it with the Dihedral group of symmetries possessed by this model.