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In a recent paper [arXiv:1806.06254], we explored the space of solutions ofthe exotic massive 3D gravity theory proposed in [arXiv:1806.04179]. We showedthat the theory admits a rich space of vacua, including asymptotically Antide-Sitter (AdS) geometries obeying different types of boundary conditions. Theexamples include black holes dressed with low decaying gravitons. Based on whathappens in other theories of massive gravity, we conjectured that suchgeometries appear on a curve of the parameter space (chiral curve) where theexotic massive gravity on AdS with sufficiently strong boundary conditionsresults dual to a 2D chiral conformal field theory. Here, we show that thisexpectation is consistent with the conserved charges and thermodynamicalproperties of the black holes of the theory, which have recently been computed[arXiv:1812.09525]. When the boundary conditions are relaxed relative to thestandard Brown-Henneaux boundary conditions, the theory exhibits solutionsconsistent with the definition of the so-called Log-gravity. The asymptoticbehavior of these solutions presents a logarithmic term in the Fefferman-Grahamexpansion that, nonetheless, is compatible with the AdS asymptotic symmetries.This long range interaction is due to a mode of the massive gravity thatbecomes massless precisely on the chiral curve. We construct exact solutionsexhibiting this behavior, which admit to be interpreted as fully backreactinggravitational waves propagating on an extremal black hole, and carryingnon-vanishing gravitational energy. We also discuss other vacua of the theory,such as Warped-AdS$_3$ black holes, gravitational waves on such backgrounds,AdS$_2\times S^1$ spaces, and black holes in dS$_3$ space.

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