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In this paper, we study the Liouville--type theorems for three--dimensionalstationary incompressible MHD and Hall--MHD systems in a slab with periodicboundary condition. We show that, under the assumptions that$(u^\theta,b^\theta)$ or $(u^r,b^r)$ is axisymmetric, or $(ru^r,rb^r)$ isbounded, any smooth bounded solution to the MHD or Hall--MHD system with localDirichlet integral growing as an arbitrary power function must be constant.This hugely improves the result of \cite[Theorem 1.2]{pan2021Liouville}, wherethe Dirichlet integral of $\mathbf{u}$ is assumed to be finite. Motivated by\cite[Bang--Gui--Wang--Xie, 2022, {\itarXiv:2205.13259}]{bang2022Liouvilletype}, our proof relies on establishingSaint--Venant's estimates associated with our problem, and the result in thecurrent paper extends that for stationary Navier--Stokes equations shown by\cite{bang2022Liouvilletype} to MHD and Hall--MHD equations. To achieve this,more intricate estimates are needed to handle the terms involving $\mathbf{b}$properly.

Liouville--type Theorems for Steady MHD and Hall--MHD Equations in $\R^2 \times \T$