  We obtain general continuous-wave (CW)\ solutions in the model of a spinor
positronium condensate in the absence of magnetic field. The CW solutions with
both in-phase ($n=0$) and out-of-phase ($n=1$) spin components exist, with
their ranges limited by the total particle density, $\rho $. In the limit of
negligible population exchange between the spin components, the CW solutions
are found to be stable or unstable, depending on the particle density of the
para positronium. Ortho positronium, in the $F=1$ spinor state, forms a
ferromagnetic condensate with stable in-phase CW solutions only. Subsequent
examination of the modulational instability (MI) is carried out both in the
limit case of identical wavenumbers in the spin components, $% \Delta k\equiv
k_{1}-k_{-1}=0$, and in the more general case of $\Delta k\neq 0$ too. The CW
solutions with $n=0$ and $1$ solutions, which are stable in the case of $\Delta
k=0$, are unstable for $\Delta k\neq 0$, for the natural repulsive sign of the
nonlinearities. The total particle density, $\rho $, in the limit of $\Delta
k=0$ is found to have a significant role for the stability of the condensate,
which is determined by the sign of the self-interaction nonlinearity.
