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In this paper we study some properties of the newly found Arnold-Beltramiflux-brane solutions to the minimal $D=7$ supergravity. To this end we firstsingle out the appropriate Free Differential Algebra containing both a gauge$3$-form $\mathbf{B}^{[3]}$ and a gauge $2$-form $\mathbf{B}^{[2]}$: then wepresent the complete rheonomic parametrization of all the generalizedcurvatures. This allows us to identify two-brane configurations withArnold-Beltrami fluxes in the transverse space with exact solutions ofsupergravity and to analyze the Killing spinor equation in their background. Wefind that there is no preserved supersymmetry if there are no additionaltranslational Killing vectors. Guided by this principle we explicitly constructArnold-Beltrami flux two-branes that preserve $0$, $1/8$ and $1/4$ of theoriginal supersymmetry. Two-branes without fluxes are instead BPS states andpreserve $1/2$ supersymmetry. For each two-brane solution we carefully studyits discrete symmetry that is always given by some appropriate crystallographicgroup $\Gamma$. Such symmetry groups $\Gamma$ are transmitted to the $D=3$gauge theories on the brane world--volume that occur in the gauge/gravitycorrespondence. Furthermore we illustrate the intriguing relation between gaugefluxes in two-brane solutions and hyperinstantons in $D=4$ topologicalsigma-models.

Minimal $D=7$ Supergravity and the supersymmetry of Arnold-Beltrami Flux branes