  We study the equilibrium phase diagram of a generalized ABC model on an
interval of the one-dimensional lattice: each site $i=1,...,N$ is occupied by a
particle of type $\a=A,B,C,$ with the average density of each particle species
$N_\a/N=r_\a$ fixed. These particles interact via a mean field
non-reflection-symmetric pair interaction. The interaction need not be
invariant under cyclic permutation of the particle species as in the standard
ABC model studied earlier. We prove in some cases and conjecture in others that
the scaled infinite system $N\rw\infty$, $i/N\rw x\in[0,1]$ has a unique
density profile $\p_\a(x)$ except for some special values of the $r_\a$ for
which the system undergoes a second order phase transition from a uniform to a
nonuniform periodic profile at a critical temperature $T_c=3\sqrt{r_A r_B
r_C}/2\pi$.
