  A new hidden symmetry is exhibited in the reflection equation and related
quantum integrable models. It is generated by a dual pair of operators
$\{\textsf{A}, \textsf{A}^*\}\in{\cal A}$ subject to $q-$deformed Dolan-Grady
relations. Using the inverse scattering method, a new family of quantum
integrable models is proposed. In the simplest case, the Hamiltonian is linear
in the fundamental generators of ${\cal A}$. For general values of $q$, the
corresponding spectral problem is quasi-exactly solvable. Several examples of
two-dimensional massive/massless (boundary) integrable models are reconsidered
in light of this approach, for which the fundamental generators of ${\cal A}$
are constructed explicitly and exact results are obtained. In particular, we
exhibit a dynamical Askey-Wilson symmetry algebra in the (boundary) sine-Gordon
model and show that asymptotic (boundary) states can be expressed in terms of
$q-$orthogonal polynomials.
