  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.
