  The ground state of Shor-Movassagh chain can be analytically described by the
Motzkin paths. There is no analytical description of the excited states, the
model is not solvable. We prove the integrability of the model without
interacting part in this paper [free Shor-Movassagh]. The Lax pair for the free
Shor-Movassagh open chain is explicitly constructed. We further obtain the
boundary $K$-matrices compatible with the integrability of the model on the
open interval. Our construction provides a direct demonstration for the quantum
integrability of the model, described by Yang-Baxter algebra. Due to the lack
of crossing unitarity, the integrable open chain can not be constructed by the
reflection equation (boundary Yang-Baxter equation).
