  We introduce a family of fourth order two-step methods that preserve the
energy function of canonical polynomial Hamiltonian systems. Each method in the
family may be viewed as a correction of a linear two-step method, where the
correction term is O(h^5) (h is the stepsize of integration). The key tools the
new methods are based upon are the line integral associated with a conservative
vector field (such as the one defined by a Hamiltonian dynamical system) and
its discretization obtained by the aid of a quadrature formula. Energy
conservation is equivalent to the requirement that the quadrature is exact,
which turns out to be always the case in the event that the Hamiltonian
function is a polynomial and the degree of precision of the quadrature formula
is high enough. The non-polynomial case is also discussed and a number of test
problems are finally presented in order to compare the behavior of the new
methods to the theoretical results.
