Three one-parameter, eighth-order families of symplectic integrators are presented. They are based on the Strang splitting, the Forest–Ruth construction, and the Zassenhaus formula, respectively. In each time-step, a free parameter is adopted to preserve energy exactly in this step. Numerical experiments show that such a procedure is possible both in double- and quad-precision computations. Therefore, our integrators can be called energy preserving symplectic integrators.