  Let $B_+$ and $B_-$ be nonnegative symmetric operators with defect numbers
$(1,1)$ in Hilbert spaces $\mathcal H_+$ and $\mathcal H_-$. We consider the
coupling $A$ of operators $A_+=B_+$ and $A_-=-B_-$ which is self-adjoint,
nonnegative and has a nonempty resolvent set in the Kre\u{\i}n space $\mathcal
K$ with the fundamental decomposition $\mathcal K=\mathcal H_+[+]\mathcal H_-$.
Such an operator is definitizable and has at most two critical points at
$\infty$ and $0$. It is similar to a self-adjoint operator in a Hilbert space
if and only if its critical points are regular and there are no Jordan chains
at $0$. We find criteria for the regularity of critical points of $A$
formulated in terms of the abstract Weyl functions $m_+$ and $m_-$ of the
operators $B_+$ and $B_-$.
  A typical example of such coupling is the operator $A$ generated in the
Kre\u{\i}n space $\mathcal K = L^2_w(I)$ by the differential expression \[
\mathfrak{a}f := - \frac{1}{w}\left( \frac{1}{r} f' \right)', \] where
$I=(b_-,b_+)$, $-\infty \leq b_- <0 < b_+ \leq + \infty$, $r, w \in
L^1_{\operatorname{loc}}(I)$, $r(t) >0$ and $(\operatorname{sgn} t) w(t) > 0$
a.e. on $I$. The operator $A$ is self-adjoint in the Kre\u{\i}n space
$L^2_w(I)$ and can be considered as the coupling of the restrictions of $A$
onto $L^2_{w_-}(I_-)$ and $L^2_{w_+}(I_+)$, where $I_- = (b_-,0)$, $I_+ =
(0,b_+)$, $w_{-}$ is the restriction of $-w$ onto $I_-$ and $w_{+}$ is the
restriction of $w$ onto $I_+$. For this operator the general criteria for the
regularity of a critical point are reformulated in terms of the coefficients
$w$ and $r$.
