Growing Hair on the extremal $BTZ$ black hole

We show that the nonlinear $\sigma-$model in an asymptotically $AdS_3$ space-time admits a novel local symmetry. The field action is assumed to be quartic in the nonlinear $\sigma-$model fields and minimally coupled to gravity. The local symmetry transformation simultaneously twists the nonlinear $\sigma-$model fields and changes the space-time metric, and it can be used to map an extremal $BTZ$ black hole to infinitely many hairy black hole solutions.