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BPS solutions of 5-dimensional supergravity correspond to certain gradientflows on the product M x N of a quaternionic-Kaehler manifold M of negativescalar curvature and a very special real manifold N of dimension n >=0. Suchgradient flows are generated by the `energy function' f = P^2, where P is a(bundle-valued) moment map associated to n+1 Killing vector fields on M. Wecalculate the Hessian of f at critical points and derive some properties of itsspectrum for general quaternionic-Kaehler manifolds. For the homogeneousquaternionic-Kaehler manifolds we prove more specific results depending on thestructure of the isotropy group. For example, we show that there always existsa Killing vector field vanishing at a point p in M such that the Hessian of fat p has split signature. This generalizes results obtained recently for thecomplex hyperbolic plane (universal hypermultiplet) in the context of5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds weshow the existence of non-degenerate local extrema of f, for appropriateKilling vector fields. On the other hand, for the non-symmetric homogeneousquaternionic-Kaehler manifolds we find degenerate local minima.Comment: 22 page

Flows on Quaternionic-K�hler and Very Special Real Manifolds