  Kondo conduction has been observed in a quantum dot with an even number of
electrons at the Triplet-Singlet degeneracy point produced by applying a small
magnetic field $B$ orthogonal to the dot plane.
  At a much larger field $ B \sim B_*$, orbital effects induce the reversed
transition from the Singlet to the Triplet state. We study the newly proposed
Kondo behavior at this point. Here the Zeeman spin splitting cannot be
neglected, what changes the nature of the Kondo coupling. On grounds of exact
diagonalization results in a dot with cylindrical symmetry, we show that, at
odds with what happens at the other crossing point, close to $B_*$, orbital and
spin degrees of freedom are ``locked together'', so that the Kondo coupling
involves a fictitious spin 1/2 only, which is fully compensated by conduction
electrons under suitable conditions. In this sense, spin at the dot is
fractionalized. We derive the scaling equation of the system by means of a
nonperturbative variational approach. The approach is extended to the $B \neq
B_*$-case and the residual magnetization on the dot is discussed.
