  The Average Vector Field (AVF) method is a B-series scheme of the second
order. As a discrete gradient method it preserves exactly the energy integral
for any canonical Hamiltonian system. We present and discuss two locally exact
and energy-preserving modifications of the AVF method: AVF-LEX (of the third
order) and AVF-SLEX (of the fourth order). Applications to spherically
symmetric potentials are given, including a compact explicit expression for the
AVF scheme for the Coulomb-Kepler problem.
