On solutions of integral equations with analytic kernels and rotations

We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where ${\displaystyle T=M_{n₁,k₁}...M_{n_m,k_m}}$ and ${\displaystyle M_{n_j,k_j}}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial ${\displaystyle P_T\left(t\right)=t³-t}$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.