  A class of scalar Stieltjes like functions is realized as linear-fractional
transformations of transfer functions of conservative systems based on a
Schr\"odinger operator T_h in $L_2[a,+\infty)$ with a non-selfadjoint boundary
condition. In particular it is shown that any Stieltjes function of this class
can be realized in the unique way so that the main operator $\bA$ of a system
is an accretive (*)-extension of a Schr\"odinger operator T_h. We derive
formulas that restore the system uniquely and allow to find the exact value of
a non-real parameter h in the definition of T_h as well as a real parameter
$\mu$ that appears in the construction of the elements of the realizing system.
An elaborate investigation of these formulas shows the dynamics of the restored
parameters h and $\mu$ in terms of the changing free term $\gamma$ from the
integral representation of the realizable function. It turns our that the
parametric equations for the restored parameter h represent different circles
whose centers and radii are determined by the realizable function. Similarly,
the behavior of the restored parameter $\mu$ are described by hyperbolas.
