We use the consistent set of hypernetted chain (HNC) equations to investigate the stability of liquid and vapor phases of simple fluids. We calculate the phase diagram and discuss the physical (spinodal) and mathematical (termination line) singularities of the theory. The latter delimits a region where the HNC equations have no physically acceptable solutions. Additionally, we confirm that the physical solutions of the HNC equation correspond to local minima of the consistent grand potential which, however, appears to be unbounded from below.