  In the present paper we investigate the set $\Sigma_J$ of all
$J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices
$<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein
space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different
types of $J$-self-adjoint extensions of $S$. One of our main results is the
equivalence between the presence of $J$-self-adjoint extensions of $S$ with
empty resolvent set and the commutation of $S$ with a Clifford algebra
${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with
$JR=-RJ$. This enables one to construct the collection of operators
$C_{\chi,\omega}$ realizing the property of stable $C$-symmetry for extensions
$A\in\Sigma_J$ directly in terms of ${\mathcal C}l_2(J,R)$ and to parameterize
the corresponding subset of extensions with stable $C$-symmetry. Such a
situation occurs naturally in many applications, here we discuss the case of an
indefinite Sturm-Liouville operator on the real line and a one dimensional
Dirac operator with point interaction.
