  For Hamiltonian systems with non-canonical structure matrices, a new family
of fourth-order energy-preserving integrators is presented. The integrators
take a form of a combination of Runge--Kutta methods and continuous-stage
Runge--Kutta methods and feature a set of free parameters that offer greater
flexibility and efficiency. Specifically, we demonstrate that by carefully
choosing these free parameters a simplified Newton iteration applied to the
integrators of order four can be parallelizable. This results in faster and
more efficient integrators compared with existing fourth-order
energy-preserving integrators.
