  The Riemann equation $u_t+uu_x=0$, which describes a one-dimensional
accelerationless perfect fluid, possesses solutions that typically develop
shocks in a finite time. This equation is $\cP\cT$ symmetric. A one-parameter
$\cP\cT$-invariant complex deformation of this equation,
$u_t-iu(iu_x)^\epsilon= 0$ ($\epsilon$ real), is solved exactly using the
method of characteristic strips, and it is shown that for real initial
conditions, shocks cannot develop unless $\epsilon$ is an odd integer.
