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After a harmonic linear normalization, the potential can be written as $V(x_1,x_2)=\\frac{1}{2}(v_1x_1^2+v_2x_2^2)+\\sum_{j+2k\\ge 3}a_{j,2k}\\,x_1^j x_2^{2k}$, with $v_1/v_2\\notin\\mathbb{Q}$. The operator can be brought into a quantum Birkhoff normal form whose Weyl symbol is a formal series $B \\equiv H_2 + \\sum_{2r+k+\\ell \\ge 2} b_{r,k,\\ell}\\, \\hbar^{2r} \\Omega_1^k \\Omega_2^\\ell$. 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