Summary: Our studies concern some aspects of scattering theory of the singular differential systems \(y'-x^{-1}Ay-q(x)y=\rho By\), \(x>0\) with \(n\times n\) matrices \(A,B, q(x), x\in(0,\infty)\), where \(A,B\) are constant and \(\rho\) is a spectral parameter. We concentrate on investigation of certain Volterra integral equations with respect to tensor-valued functions. The solutions of these integral equations play a central role in construction of the so-called Weyl-type solutions for the original differential system. Actually, the integral equations provide a method for investigation of the analytical and asymptotical properties of the Weyl-type solutions while the classical methods fail because of the presence of the singularity. In the paper, we consider the important special case when \(q\) is smooth and \(q(0)=0\) and obtain the classical-type asymptotical expansions for the solutions of the considered integral equations as \(\rho\to\infty\) with \(o\left(\rho^{-1}\right)\) rate remainder estimate. The result allows one to obtain analogous asymptotics for the Weyl-type solutions that play in turn an important role in the inverse scattering theory.