Gauge invariant counterparts and quantization of systems under holonomic   constraints
  Systems under holonomic constraints are classified within the generalized Hamiltonian framework as second-class constraints systems. We show that each system of point particles with holonomic constraints has a hidden gauge symmetry which allows to quantize it in the original phase space as a first-class constraints system. The proposed method is illustrated with quantization of a point particle moving on an $n-1$-dimensional sphere $S^{n-1}$ as well as its field theory analog the O(n) nonlinear sigma model. 
M. I. Krivoruchenko, Amand Faessler, A. A. Raduta, C. Fuchs