  We initiate a systematic study of integrable models for spin chains with
constrained Hilbert spaces; we focus on spin-1/2 chains with the Rydberg
constraint. We extend earlier results for medium-range spin chains to the
constrained Hilbert space, and formulate an integrability condition. This
enables us to construct new integrable models with fixed interaction ranges. We
classify all time- and space-reflection symmetric integrable
Rydberg-constrained Hamiltonians of range 3 and 4. At range 3, we find a single
family of integrable Hamiltonians: the so-called RSOS quantum chains, which are
related to the well-known RSOS models of Andrews, Baxter, and Forrester. At
range 4 we find two families of models, the first of which is the constrained
XXZ model. We also find a new family of models depending on a single coupling
$z$. We provide evidence of two critical points related to the golden ratio
$\phi$, at $z=\phi^{-1/2}$ and $z=\phi^{3/2}$. We also perform a partial
classification of integrable Hamiltonians for range 5.
