  This is the first in a series of papers devoted to the study of spin chains
capturing the spectral problem of 4d $\mathcal{N}=2$ SCFTs in the planar limit.
At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry
algebra, defined by the $R$-matrix read off from the superpotential. This
implies that when orbifolding the $\mathcal{N}=4$ symmetry algebra down to the
$\mathcal{N}=2$ one and then marginaly deforming, the broken generators are not
lost, but get upgraded to quantum generators. Importantly, we demonstrate that
these chains are dynamical, in the sense that their Hamiltonian depends on a
parameter which is dynamically determined along the chain. At one loop we map
the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which
corresponds to an RSOS model, whose adjacency graph can be read off from the
gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by
an alternating nearest-neighbour Hamiltonian, while another choice of SU(2)
sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have
a common vacuum state, around which the magnon dispersion relations are
naturally uniformised by elliptic functions. Concretely, for the $\mathbb{Z}_2$
quiver theory we study these dynamical chains by solving the one- and
two-magnon problems with the coordinate Bethe ansatz approach. We confirm our
analytic results by numerical comparison with the explicit diagonalisation of
the Hamiltonian for short closed chains.
