  In this paper we study the Metropolis algorithm in connection with two
mean--field spin systems, the so called mean--field Ising model and the
Blume--Emery--Griffiths model. In both this examples the naive choice of
proposal chain gives rise, for some parameters, to a slowly mixing Metropolis
chain, that is a chain whose spectral gap decreases exponentially fast (in the
dimension $N$ of the problem). Here we show how a slight variant in the
proposal chain can avoid this problem, keeping the mean computational cost
similar to the cost of the usual Metropolis. More precisely we prove that, with
a suitable variant in the proposal, the Metropolis chain has a spectral gap
which decreases polynomially in 1/N. Using some symmetry structure of the
energy, the method rests on allowing appropriate jumps within the energy level
of the starting state.
