  The three simultaneous algebraic equations, $C^2=1$, $[C,PT]=0$, $[C,H]=0$,
which determine the $C$ operator for a non-Hermitian $PT$-symmetric Hamiltonian
$H$, are shown to have a nonunique solution. Specifically, the $C$ operator for
the Hamiltonian $H={1/2}p^2+{1/2}\mu^2q^2+i\epsilon q^3$ is determined
perturbatively to first order in $\epsilon$ and it is demonstrated that the $C$
operator contains an infinite number of arbitrary parameters. For each
different $C$ operator, the corresponding equivalent isospectral
Dirac-Hermitian Hamiltonian $h$ is calculated.
