  In a recent paper the first and the third authors introduced the notion of
horizontal \alpha-harmonic map, with respect to a given C^1 planes distribution
P_T on all R^m. The goal of this paper is to investigate compactness and
quantization properties of sequences of horizontal 1/2- harmonic maps u_k in
1D. The quantization analysis is obtained through a precise asymptotic
development of the energy of u_k in the neck regions and a subtle application
of new Pohozaev-type formulae.
