  By combining a standard symmetric, symplectic integrator with a new step size
controller, we provide an integration scheme that is symmetric, reversible and
conserves the values of the constants of motion. This new scheme is appropriate
for long term numerical integrations of geodesic orbits in spacetime
backgrounds, whose corresponding Hamiltonian system is non-integrable, and, in
general, for any non-integrable Hamiltonian system whose kinetic part depends
on the position variables. We show by numerical examples that the new
integrator is faster and more accurate i) than the standard symplectic
integration schemes with or without standard adaptive step size controllers and
ii) than an adaptive step Runge-Kutta scheme.
