  The long-time kinetics of the spherical model in an external magnetic field
and below the equilibrium critical temperature is studied. The solution of the
associated stochastic Langevin equation is reduced exactly to a single
non-linear Volterra equation. For a sufficiently small external field, the
kinetics of the magnetization-reversal transition from the metastable to the
ground state is compared to the ageing behaviour of coarsening systems quenched
into the low-temperature phase. For an oscillating magnetic field and below the
critical temperature, we find evidence for the absence of the
frequency-dependent dynamic phase transition, which was observed previously to
occur in Ising-like systems.
