  We present a mean-field calculation of the phase diagram of a simple model of
localized moments, in the hexagonal uranium heavy-fermion compounds. The model
considers a non-Kramers quadrupolar doublet ground state magnetically coupled
with a singlet excited-state, favoring in-plane van-Vleck magnetism, as has
been conjectured for UPt$_3$. The Hamiltonian which defines the model is
Heisenberg like in both, magnetic and quadrupolar moments. Among our main
results are: (i) for zero intersite quadrupolar coupling, the magnetic order is
achieved by a first order transition above a critical intersite magnetic
coupling value which becomes second order at higher coupling strengths. (ii)
for finite intersite quadrupolar coupling, at temperatures below a second order
quadrupolar ordering transition, the minimal magnetic coupling value is
increased but (a) the magnetic ordering temperature is enhanced above this
value, and (b) the ordering of first and second order transitions in the phase
diagram is reversed.
