The 16th Hilbert problem restricted to circular algebraic limit cycles

We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S   admits at most S−1S−1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S   has at most S−1S−1 algebraic limit cycles given by circles, and this number is reached.