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)xn3x′=a1A1(t)xn1+a2A2(t)xn2+a3A3(t)xn3, where n1,n2,n3>1n1,n2,n3>1 are distinct integers, a1,a2,a3∈Ra1,a2,a3∈R, and A1,A2,A3A1,A2,A3 are 2π2π-periodic analytic functions such that A1(t)sint,A2(t)cost,A3(t)sintcostA1(t)sint,A2(t)cost,A3(t)sintcost areππ-periodic positive even functions.When (n3−n1)(n3−n2)<0(n3−n1)(n3−n2)<0 we prove that the equation has no non-trivial (different from zero) limit cycle for any value of the parameters a1,a2,a3a1,a2,a3.When (n3−n1)(n3−n2)>0(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.