The authors consider the differential operator \[ -D\omega(\cdot,z)D+q \tag{1} \] in \(L^2(0,\infty)\) whose leading coefficient contains the eigenvalue parameter \(z\) and \(q\in L^1_ {\text{loc}}[0,\infty)\). It is supposed that \(\omega(\cdot,z)\) has the particular form \(\omega(t,z)=p(t)+c^2(t)/(z-r(t))\), \(z\in\mathbb{C}\setminus\mathbb{R}\), where \(p(t),c(t),r(t)\) are such that \(1/\omega(\cdot,z)\in L^1_{\text{loc}}[0,\infty)\) for all \(z\in\mathbb{C}\setminus\mathbb{R}\). The limit-point/limit-circle alternative for (1) on the half line and its implications for the associated linear system of differential equations, where the eigenvalue parameter appears linearly, are investigated. In particular, in the limit-point case it is shown that under mild conditions, the Titchmarsh-Weyl coefficient for all but one selfadjoint extension belongs to the so-called Kac subclass of Nevanlinna functions, and the exceptional case corresponds to the generalized Friedrichs extension.