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The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations:\n  (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow;\n  (ii) the undamped region contains a horizontal strip;\n  (iii) the undamped part is a line. 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