Symplectic integration for the collisional gravitational $N$-body
  problem
  We present a new symplectic integrator designed for collisional gravitational
$N$-body problems which makes use of Kepler solvers. The integrator is also
reversible and conserves 9 integrals of motion of the $N$-body problem to
machine precision. The integrator is second order, but the order can easily be
increased by the method of \citeauthor{yos90}. We use fixed time step in all
tests studied in this paper to ensure preservation of symplecticity. We study
small $N$ collisional problems and perform comparisons with typically used
integrators. In particular, we find comparable or better performance when
compared to the 4th order Hermite method and much better performance than
adaptive time step symplectic integrators introduced previously. We find better
performance compared to SAKURA, a non-symplectic, non-time-reversible
integrator based on a different two-body decomposition of the $N$-body problem.
The integrator is a promising tool in collisional gravitational dynamics.
