  A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a
time-transformed separable Hamiltonian (or time-transformed symplectic
integrator; TSI) has been known to conserve the Kepler orbit exactly. We find
that for an elliptic orbit, such feature appears for an arbitrary step size.
But it is not the case for a hyperbolic orbit: when the half step size is
larger than the conjugate momenta of the mean anomaly, a phase transition
happens and the new position jumps to the nonphysical counterpart of the
hyperbolic trajectory. Once it happens, the energy conservation is broken.
Instead, the kinetic energy minus the potential energy becomes a new conserved
quantity. We provide a mathematical explanation for such phenomenon. Our result
provides a deeper understanding of the TSI method, and a useful constraint of
the step size when the TSI method is used to solve the hyperbolic encounters.
This is particular important when an (Bulirsch-Stoer) extrapolation integrator
is used together, which requires the convergence of integration errors.
