This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian $PT$-symmetric Hamiltonians $H=p^2+ x^2(ix)^varepsilon$ ($varepsilongeq0$). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken $PT$-symmetric classical trajectories, and a remarkable topological transition at $varepsilon=2$. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.
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