  $J$-self-adjoint extensions of the Phillips symmetric operator $S$ are
studied. The concepts of stable and unstable $C$-symmetry are introduced in the
extension theory framework. The main results are the following: if ${A}$ is a
$J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or
$\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable
$C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no
$J$-self-adjoint extensions of the Phillips operator with unstable
$C$-symmetry.
