  We study a generalization of the Kondo model in which the impurity spin is
represented by Abrikosov fermions in a rotation group SU(P) larger than the
SU(N) group associated to the spin of the conduction electrons, thereby forcing
the single electronic bath to underscreen the localized moment. We demonstrate
how to formulate a controlled large N limit preserving the property of
underscreening, and which can be seen as a ``dual'' theory of the multichannel
large N equations usually associated to overscreening. Due to the anomalous
scattering on the uncompensated degrees of freedom, the Fermi liquid
description of the electronic fluid is invalidated, with the logarithmic
singularities known to occur in the S=1 SU(2) Kondo impurity model being
replaced by continuous power laws at N=\infty. The present technique can be
extended to tackle the related underscreened Kondo lattice model in the large N
limit. We discover the occurence of an insulating pseudogap regime in place of
the expected renormalized metallic phase of the fully screened case, preventing
the establishement of coherence over the lattice. This work and the recent
observation of a similar weakly insulating behavior on transport in CeCuAs_2
should give momentum for further studies of underscreened impurity models on
the lattice.
