Does the complex deformation of the Riemann equation exhibit shocks?

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 {\cal P}{\cal T} symmetric. A one-parameter {\cal P}{\cal T} -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. When epsilon is an odd integer, the shock-formation time is calculated explicitly.