  We show that the $AdS_5 \times L^{a,b,c}$ solution in type IIB theory is
non-integrable. To do so, we consider a string embedding and study its
fluctuations which do not admit Liouville integrable solutions. We, also,
perform a numerical analysis to study the time evolution of the string and
compute the largest Lyapunov exponent. This analysis indicates that the string
motion is chaotic. Finally, we consider the point-like limit of the string that
corresponds to BPS mesons of the quiver theory.
