  We calculate the average number of critical points $\overline{\mathcal{N}}$
of the energy landscape of a many-body system with disordered two-body
interactions and a weak on-site potential. We find that introducing a weak
nonlinear on-site potential dramatically increases $\overline{\mathcal{N}}$ to
exponential in system size and give a complete picture of the organization of
critical points. Our results extend solvable spin-glass models to physically
more realistic models and are of relevance to glassy systems, nonlinear
oscillator networks and many-body interacting systems.
