  In this paper, we develop a higher order symmetric partitioned Runge-Kutta
method for a coupled system of differential equations on Lie groups. We start
with a discussion on partitioned Runge-Kutta methods on Lie groups of arbitrary
order. As symmetry is not met for higher orders, we generalize the method to a
symmetric partitioned Runge-Kutta (SPRK) scheme. Furthermore, we derive a set
of coefficients for convergence order 4. The SPRK integration method can be
used, for example, in simulations of quantum field theories. Finally, we
compare the new SPRK scheme numerically with the St\"ormer-Verlet scheme, one
of the state-of-the-art schemes used in this subject.
