Dirac reduction of dual Poisson-presymplectic pairs
  A new notion of a dual Poisson-presymplectic pair is introduced and its properties are examined. The procedure of Dirac reduction of Poisson operators onto submanifolds proposed by Dirac is in this paper embedded in a geometric procedure of reduction of dual Poisson-presymplectic pairs. The method presented generalizes those used by Marsden and Ratiu for reductions of Poisson manifolds. Two examples are given. 
Maciej Blaszak and Krzysztof Marciniak