  A new family of methods involving complex coefficients for the numerical
integration of differential equations is presented and analyzed. They are
constructed as linear combinations of symmetric-conjugate compositions obtained
from a basic time-symmetric integrator of order 2n (n $\ge$ 1). The new
integrators are of order 2(n + k), k = 1, 2, ..., and preserve time-symmetry up
to order 4n + 3 when applied to differential equations with real vector fields.
If in addition the system is Hamiltonian and the basic scheme is symplectic,
then they also preserve symplecticity up to order 4n + 3. We show that these
integrators are well suited for a parallel implementation, thus improving their
efficiency. Methods up to order 10 based on a 4th-order integrator are built
and tested in comparison with other standard procedures to increase the order
of a basic scheme.
