Existence of non-trivial limit cycles in Abel equations with symmetries

We study the periodic solutions of the generalized Abel equation x′ = a1A1(t)xn1 +a2A2(t)xn2 +a3A3(t)xn3, where n1, n2, n3 >1 are distinct integers, a1, a2, a3 ∈ R, and A1, A2, A3 are 2π-periodic analytic functions such that A1(t) sin t, A2(t) cos t, A3(t) sin t cos t are π-periodic positive even functions. When (n3 −n1)(n3 −n2) 0 we obtain under additional conditions the existence of non-trivial limit cycles. In particular, we obtain limit cycles not detected by Abelian integrals.