On (non)integrability of classical strings in p-brane backgrounds

We investigate the question of possible integrability of classical string motion in curved p-brane backgrounds. For example, the D3-brane metric interpolates between the flat and the AdS_5 x S^5 regions in which string propagation is integrable. We find that while the point-like string (geodesic) equations are integrable, the equations describing an extended string in the complete D3-brane geometry are not. The same conclusion is reached for similar brane intersection backgrounds interpolating between flat space and AdS_k x S^k. We consider, in particular, the case of the NS 5-brane-fundamental string background. To demonstrate non-integrability we make a special "pulsating string" ansatz for which the string equations reduce to an effective one-dimensional system. Expanding near this simple solution leads to a linear differential equation for small fluctuations that cannot be solved in quadratures, implying non-integrability of the original set of string equations.