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In this paper we consider the structure of the $AdS_3$ vacua in $R^3$expansion of the New Massive Gravity ($R^3$-NMG). We obtain the degeneracies ofthe $AdS_3$ vacua at several points of the parametric space. Additionally,following a specific analysis we show that $AdS_3$ wave solutions are present.Using these wave solutions, we single out two special points of the parametricspace for which logarithmic terms appear in the solutions. The first one is apoint at which the effective mass of the wave profile which is interpreted as ascalar mode, completely saturates the Breitenlohner-Freedman bound of the$AdS_3$ space in which the wave is propagating. The second special point is apoint at which the central charge of the theory vanishes. Furthermore, weinvestigate the possibility of asymptotically Lifshitz black solutions to bepresent in the three-dimensional $R^3$-NMG. We derive analytically the Lifshitzvacua considering specific relations between the mass parameters of $R^3$-NMG.A certain polynomial equation arises at the first special point where solutionswith logarithmic fall-off in the $AdS_3$ space appear. Solving this polynomialequation, we obtain the values of the dynamical exponent $z$ which correspondto possible asymptotically Lifshitz black hole solutions. However, it is shownthat asymptotically Lifshitz black solutions do not exist in thethree-dimensional $R^3$-NMG for a specific ansatz of the black hole metric.

Searching for AdS_3 waves and Asymptotically Lifshitz black holes in R^3-NMG