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We show that the string worldsheet theory of Gaiotto-Maldacena holographicduals to N=2 superconformal field theories generically fails to be classicallyintegrable. We demonstrate numerically that the dynamics of a winding stringconfiguration possesses a non-vanishing Lyapunov exponent. Furthermore ananalytic study of the Normal Variational Equation fails to yield a Liouvilliansolution. An exception to the generic non-integrability of such backgrounds isprovided by the non-Abelian T-dual of $AdS_5 \times S^5$. Here by virtue of thecanonical transformation nature of the T-duality classical integrability isknown to be present.

Integrability and non-Integrability in N=2 SCFTs and their Holographic Backgrounds