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

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