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This paper concerns the existence of generalized solitary waves (solitary waves with small ripples at infinity) for a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. It is proved that the FPUT lattice problem has a generalized solitary-wave solution with the amplitude of those ripples algebraically small using dynamical system approach. The problem is first formulated as a dynamical system problem and then the center manifold reduction theorem with Laurent series expansion is applied to show that this system can be reduced to a system of ordinary differential equations with dimension five. Its dominant system has a homoclinic solution. By applying a perturbation method and adjusting some appropriate constants, it is shown that this homoclinic solution persists for the original dynamical system, which connects to a periodic solution of algebraically small amplitude at infinity (called generalized homoclinic solution), which yields the existence of a generalized solitary wave for the FPUT lattice. The result presented here with the algebraic smallness of those ripples will be needed to show the existence of generalized multi-hump waves for the FPUT lattice later.