Lax Equations, Singularities and Riemann–Hilbert Problems

The existence of singularities of the solution for a class of Lax equations is investigated using a development of the factorization method first proposed by Semenov-Tyan-Shanskiĭ (Funct Anal Appl 17(4):259–272, 1983) and Reymann and Semenov-Tian-Shansky (1994). It is shown that the existence of a singularity at a point t = ti is directly related to the property that the kernel of a certain Toeplitz operator (whose symbol depends on t) be non-trivial. The investigation of this question involves the factorization on a Riemann surface of a scalar function closely related to the above-mentioned operator. Two examples are presented which show different aspects of the problem of computing the set of singularities of the solution to the system considered. The relation between the Riemann surfaces of the classical and Lax formulation is also considered.