  In this paper, we make a generalization of Routh's reduction method for
Lagrangian systems with symmetry to the case where not any regularity condition
is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh
equations can be obtained from the Hamilton-Pontryagin principle, by making use
of an anholonomic frame, and how these equations can be reduced. To do this, we
keep the momentum constraint implicit throughout and we make use of a Routhian
function defined on a certain submanifold of the Pontryagin bundle. Then, we
show how the reduced implicit Lagrange-Routh equations can be described in the
context of dynamical systems associated to Dirac structures, in which we fully
utilize a symmetry reduction procedure for implicit Hamiltonian systems with
symmetry.
