In this manuscript we study Liouvillian non-integrability of strings in $AdS_{6}times S^{2}timesSigma$ background and its 5D Holographic Duals. For this we consider soliton strings and look for simple solutions in order to reduce the equations to only one linear second order differential equation called NVE (Normal Variation Equation ). We show that, differently of previous studies, the correct truncation is given by $eta=0$ and not $sigma=0$. With this 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, and therefore the respective field theory duals, are not integrable. Finally, we consider the general case at the boundary $eta=0$ 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 stablish 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 for the background studied.
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