“…After performing the holographic renormalization of the action and the symplectic structure in Bondi gauge using covariant counter-terms [27,53,85,86,98] and corner terms [26,98], we obtain the gravitational charges associated with the solution space derived in [42,43] that includes fluctuating boundary structure. Again, we find the appropriate field-dependent redefinition of the parameters to render the charges integrable [12,30,66,[89][90][91]. In asymptotically locally AdS 3 spacetimes (respectively asymptotically locally flat spacetimes), the asymptotic symmetry algebra is given by a Lie algebroid [99] with a one-dimensional base space parametrized by the time u on the boundary and with a Diff(S 1 )⊕Diff(S 1 ) (respectively BMS 3 ) algebra at each value of u.…”