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In this paper, we study Whittaker $\\bar{S}_2$-modules that are locally finite\n  over $\\text{span}\\{\\frac{\\partial}{\\partial t_1}, \\frac{\\partial}{\\partial t_2}\\}$. We first show that each block $\\Omega^{\\widetilde{S}_2}_{\\mathbf{a}}$ of the category of $(A_2, \\bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra $\\bar{S}_2^{\\geq 0}$. 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