  We implement and investigate the numerical properties of a new family of
integrators connecting both variants of the symplectic Euler schemes, and
including an alternative to the classical symplectic mid-point scheme, with
some additional terms. This family is derived from a new method, introduced in
a previous study, for generating symplectic integrators based on the concept of
special symplectic manifold. The use of symplectic rotations and a particular
type of projection keeps the whole procedure within the symplectic framework.
  We show that it is possible to define a set of parameters that control the
additional terms providing a way of "tuning" these new symplectic schemes. We
test the "tuned" symplectic integrators with the perturbed pendulum and we
compare its behavior with an explicit scheme for perturbed systems. Remarkably,
for the given examples, the error in the energy integral can be reduced
considerably. There is a natural geometrical explanation, sketched at the end
of this paper. This is the subject of a parallel article where a finer analysis
is performed. Numerical results obtained in this paper open a new point of view
on symplectic integrators and Hamiltonian error.
