Polynomial differential equations with many real ovals in the same algebraic complex solution

Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F 2 FolR(2, d) | F has a non-constant rational first integral}. Given F 2 IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of the generic level (G = c). Let O(d) = sup{O(F) | F 2 IntR(2, d)}. The main purpose of this paper is to prove that O(d) = +1 for all d _ 5.
Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F € FolR (2, d) | F has non-constant rational first integral}. Given F € IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of thegeneric level (G = c). Let O(d) = sup{O(F) | F € IntR(2, d)}.The main purpose of this paper is to prove that O(d) = +∞ for all d ≥ 5.