  A new methodology is developed to integrate numerically the equations of
motion for classical many-body systems in molecular dynamics simulations. Its
distinguishable feature is the possibility to preserve, independently on the
size of the time step, all the conservation laws inherent in the description
without breaking the time reversibility. As a result, an implicit second-order
algorithm is derived and applied to pure as well as spin liquids for which the
dynamics is characterized by the conservation of total energy, linear and
angular momenta as well as magnetization and individual spin lengths. It is
demonstrated on the basis of Lennard-Jones and Heisenberg fluid models that
when such quantities as energy and magnetization must be conserved perfectly,
the new algorithm turns out to be more efficient than popular decomposition
integrators and standard predictor-corrector schemes.
