  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 the important special case when
$q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such
$q(\cdot)$ in the form of some special contour integral, where the kernel can
be written in terms of the Weyl - type solutions of the considered differential
system. Formulas of such a type play an important role in constructive solution
of inverse scattering problems: use of such formulas, where the terms in their
right-hand sides are previously found from the so-called main equation,
provides a final step of the solution procedure. In order to obtain the
above-mentioned reconstruction formula we establish first the asymptotical
expansions for the Weyl - type solutions as $\rho\to\infty$ with
$o\left(\rho^{-1}\right)$ rate remainder estimate.
