Non-Integrability of Strings in $AdS_{6}\times S^{2}\times\Sigma$ Background and its 5D Holographic Duals

In this manuscript we study Liouvillian non-integrability of strings in $AdS_{6}\times S^{2}\times\Sigma$ background. We consider soliton strings and look for simple solutions in order to reduce the equations to only one linear second order differential equation called Normal Variation Equation (NVE). We study truncations in $\eta$ and $\sigma$ variables showing their applicability or not to catch (non) integrability of models. With this technique we are able to study many recent cases considered in the literature: the abelian and non-abelian T-duals, the $(p,q)$-5-brane system, the T$_{N}$, $+_{MN}$ theories and the $\tilde{T}_{N,P}$ and $+_{P,N}$ quivers. We show that all of them are not integrable. Finally, we consider the general case at the boundary $\sigma=\sigma_0$ for large $\sigma_0$ and show that we can get general conclusions about integrability. For example, beyond the above quivers, we show generically that long quivers are not integrable. In order to establish the results, we numerically study the string dynamical system seeking by chaotic behaviour. Such a characteristic gives one more piece of evidence for non-integrability of the background studied.