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Positive energy and non-SUSY flows in ISO(7) gauged supergravity

We consider maximal gauged supergravity in 4D with ISO(7) gauge group, which arises from a consistent truncation of massive IIA supergravity on a six-sphere. Within its G$_2$ invariant sector, the theory is known to possess a supersymmetric AdS extremum as well as two non-supersymmetric ones. In this context we provide a first order formulation of the theory by making use of the Hamilton-Jacobi (HJ) formalism. This allows us to derive a positive energy theorem for both non-supersymmetric extrema. Subsequently, we also find novel non-supersymmetric domain walls (DW) interpolating between the supersymmetric extremum and each of the other two. Finally, we discuss a perturbative HJ technique that may be used in order to solve for curved DW geometries.