  The symmetric orbifold of $\mathbb{T}^4$ is exactly dual to string theory on
$\mathrm{AdS}_3\times \mathrm{S}^3 \times \mathbb{T}^4$ with minimal ($k=1$)
NS-NS flux. In this paper we study the perturbation of the symmetric orbifold
that is dual to switching on R-R flux, and hence to deforming the theory away
from the tensionless point. More specifically, we determine systematically the
action of a centrally extended supersymmetry algebra on the CFT states, and
deduce from it the anomalous conformal dimensions. In the $w$-twisted sector
with large $w$ the structure is similar to what was found for $\mathcal{N}=4$
SYM: the basic excitations are multi-magnons whose individual dispersion
relation is fixed by symmetry, and the comparison with the BMN answer suggests
that the result is true to all orders in perturbation theory. Finally we show
that the multi-magnon states interact via an integrable $S$-matrix and possess
a natural family of bound states.
