Donoghue $m$-functions for Singular Sturm–Liouville operators

Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker ((\dot A)^* - i I_{\mathcal{H}})$, $\dim (\mathcal{N}_i)=k\in \mathbb{N} \cup \{\infty\}$, the associated deficiency subspace of $\dot A$. If $A$ denotes a selfadjoint extension of $\dot A$ in $\mathcal{H}$, the Donoghue $m$-operator $M_{A,\mathcal{N}_i}^{Do} (\cdot)$ in $\mathcal{N}_i$ associated with the pair $(A,\mathcal{N}_i)$ is given by $ M_{A,\mathcal{N}_i}^{Do}(z)=zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i} \vert_{\mathcal{N}_i} , z\in \mathbb{C}\setminus \mathbb{R}, $ with $I_{\mathcal{N}_i}$ the identity operator in $\mathcal{N}_i$, and $P_{\mathcal{N}_i}$ the orthogonal projection in $\mathcal{H}$ onto $\mathcal{N}_i$.
Assuming the standard local integrability hypotheses on the coefficients $p, q,r$, we study all selfadjoint realizations corresponding to the differential expression $ \tau=\frac{1}{r(x)}[-\frac{d}{dx}p(x)\frac{d}{dx} + q(x)]$ for a.e. $x\in(a,b) \subseteq \mathbb{R}$, in $L^2((a,b); rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (respectively, $(2 \times 2)$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.