  In this papers, we couple the parareal algorithm with projection methods of
the trajectory on a specific manifold, defined by the preservation of some
conserved quantities of the differential equations. First, projection methods
are introduced as the coarse and fine propagators. Second, we also apply the
projection methods for systems with conserved quantities in the correction step
of original parareal algorithm. Finally, three numerical experiments are
performed by different kinds of algorithms to show the property of convergence
in iteration, and preservation in conserved quantities of model systems.
