On the uniqueness of limit cycles of the equation Q 2(x, y)dy = P 2(x, y)dx

We show that if the equation Q 2(x, y)dy = P 2(x, y)dx, where P 2(x, y) and Q 2(x, y) are polynomials of second degree in x and y, is symmetric around the origin [i.e., P 2(x, y) × Q 2(−x, −y) = P 2(−x, −y)Q 2(x, y)] and can be reduced to the form (1 + xy)dy = N 2(x, y)dx, whereN 2(x, y) is a polynomial of second degree in x and y, then the equation can have at most two limit cycles. The set of parameter values for which the equation cannot be reduced to the above-mentioned form has smaller dimension than the entire space of parameters of the original equation.