  Recently, a whole class of evergy-preserving integrators has been derived for
the numerical solution of Hamiltonian problems. In the mainstream of this
research, we have defined a new family of symplectic integrators depending on a
real parameter. When it is zero, the corresponding method in the family becomes
the classical Gauss collocation formula of order 2s, where s denotes the number
of the internal stages. For any given non-null value of the parameter, the
corresponding method remains symplectic and has order 2s-2: hence it may be
interpreted as a symplectic perturbation of the Gauss method. Under suitable
assumptions, it can be shown that the parameter a may be properly tuned, at
each step of the integration procedure, so as to guarantee energy conservation
in the numerical solution. The resulting method shares the same order 2s as the
generating Gauss formula, and is able to preserve both energy and quadratic
invariants.
