Shor-Movassagh chain leads to unusual integrable model

The ground state of the 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 the Yang-Baxter algebra. Because the partial transpose of the R matrix is not invertible, the model does not have crossing unitarity and the integrable open chain cannot be constructed by the reflection equation (boundary Yang-Baxter equation).