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Solution of the $1 : {โ}2$ resonant center problem in the quadratic case
The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x + A_1x^2 + B_1xy + Cy^2) โ_x+(-2y + Dx^2 + A_2xy + B_