A family of isochronous foci with darboux first integral

We consider the class of polynomial differential equations ẋ = λx - y + Pn(x, y) + P2n-1(x, y), ẏ = x + λy + Qn(x, y) + Q2n-1(x, y) with n ≥ 2, where Pi and Qi, are homogeneous polynomials of degree i. These systems have a focus at the origin if λ ≠ 0, and have either a center or a focus if λ = 0. Inside this class we identify a new subclass of Darboux integrable systems having either a focus or a center at the origin. Under generic conditions such Darboux integrable systems can have at most two limit cycles, and when they exist are algebraic. For the case n = 2 and n = 3 we present new classes of Darboux integrable systems having a focus.