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We introduce a new set of boundary conditions for three-dimensional higherspin gravity with gauge group $SL(3,\mathbb{R})\times SL(3,\mathbb{R})$, whereits dynamics at the boundary is described by the members of the modifiedBoussinesq integrable hierarchy. In the asymptotic region the gauge fields arewritten in the diagonal gauge, where the excitations go along the generators ofthe Cartan subalgebra of $sl(3,\mathbb{R})\oplus sl(3,\mathbb{R})$. We showthat the entire integrable structure of the modified Boussinesq hierarchy,i.e., the phase space, the Poisson brackets and the infinite number ofcommuting conserved charges, are obtained from the asymptotic structure of thehigher spin theory. Furthermore, its known relation with the Boussinesqhierarchy is inherited from our analysis once the asymptotic conditions arere-expressed in the highest weight gauge. Hence, the Miura map is recoveredfrom a purely geometric construction in the bulk. Black holes that fit withinour boundary conditions, the Hamiltonian reduction at the boundary, and thegeneralization to higher spin gravity with gauge group $SL(N,\mathbb{R})\timesSL(N,\mathbb{R})$ are also discussed.

Integrable systems and the boundary dynamics of higher spin gravity on AdS$_3$