We investigate the potential energy surface of a phi^4 model with infinite
range interactions. All stationary points can be uniquely characterized by
three real numbers $\alpha_+, alpha_0, alpha_- with alpha_+ + alpha_0 + alpha_-
= 1, provided that the interaction strength mu is smaller than a critical
value. The saddle index n_s is equal to alpha_0 and its distribution function
has a maximum at n_s^max = 1/3. The density p(e) of stationary points with
energy per particle e, as well as the Euler characteristic chi(e), are singular
at a critical energy e_c(mu), if the external field H is zero. However, e_c(mu)
\neq upsilon_c(mu), where upsilon_c(mu) is the mean potential energy per
particle at the thermodynamic phase transition point T_c. This proves that
previous claims that the topological and thermodynamic transition points
coincide is not valid, in general. Both types of singularities disappear for H
\neq 0. The average saddle index bar{n}_s as function of e decreases
monotonically with e and vanishes at the ground state energy, only. In
contrast, the saddle index n_s as function of the average energy bar{e}(n_s) is
given by n_s(bar{e}) = 1+4bar{e} (for H=0) that vanishes at bar{e} = -1/4 >
upsilon_0, the ground state energy.Comment: 9 PR pages, 6 figure