  The asymptotic structure of three-dimensional higher-spin anti-de Sitter
gravity is analyzed in the metric approach, in which the fields are described
by completely symmetric tensors and the dynamics is determined by the standard
Einstein-Fronsdal action improved by higher order terms that secure gauge
invariance. Precise boundary conditions are given on the fields. The asymptotic
symmetries are computed and shown to form a non-linear W-algebra, in complete
agreement with what was found in the Chern-Simons formulation. The W-symmetry
generators are two-dimensional traceless and divergenceless rank-s symmetric
tensor densities of weight s (s = 2, 3, ...), while asymptotic symmetries
emerge at infinity through the conformal Killing vector and conformal Killing
tensor equations on the two-dimensional boundary, the solution space of which
is infinite-dimensional. For definiteness, only the spin 3 and spin 4 cases are
considered, but these illustrate the features of the general case: emergence of
the W-extended conformal structure, importance of the improvement terms in the
action that maintain gauge invariance, necessity of the higher spin gauge
transformations of the metric, role of field redefinitions.
