On connection of one class of one-dimensional pseudodifferential operators with singular integral operators

The paper discusses a homogeneous one-dimensional pseudodifferential equation with a symbol of the form
$$A(x,\xi)=A_0(\xi)+\displaystyle\sum_{k=1}^N\tan\dfrac{\pi}{\alpha}\left(x-\lambda_k+i\dfrac{\alpha\beta}{2}\right)A_k(\xi) ~\ \ (x,\xi, ~\lambda_k\in \mathbb{R}, \alpha>0, ~-1<\beta<1, ~k=1,2,\dots,N),$$
where $A_k(\xi)~~ (k=0,1,\dots,N)$ are locally integrable functions from class of symbols of non-negative order $r$.
The method of bringing the pseudodifferential equation to a system of onedimensional singular integral equations with Cauchy’s kernel is proposed.