An elliptic generalization of the $D_2^{(2)}$ Spin chain

An integrable elliptic generalization of the closed $D_2^{(2)}$ quantum spin chain is constructed. Taking the trigonometric limit, we have shown that the corresponding R-matrix reduces to that of the $D_2^{(2)}$ one, up to a similarity transformation. Basing on the very fact of the factorization for the R-matrix and the corresponding transfer matrix which indeed ensures the integrability of the model, we give the exact solution of the model which can be expressed in terms of a homogeneous T-Q relation.