Let us fix trigonometric monomials $A_k$ and integers $n_kgeq 1$, $k=1,ldots,m$, and consider the family of Abel-like differential equations
$$x'=sum_{k=1}^m a_k A_k(t) x^{n_k},$$ where $a_kinmathbb{R}$.
This equation always has the trivial solution $x(t)equiv 0$. Moreover, either every bounded solution is $2pi$-periodic or $2pi$-periodic solutions are isolated. In the first case, we say that the equation has a center and in the second case, we call limit cycle to any $2pi$-periodic solution.
We are interested in studying whether there exist equations of the family with non-trivial limit cycles. That is, if there exist $a_1,ldots,a_m$ such that the differential equation has a limit cycle different from $x(t)=0$. We will focus on the special case ${n_kcolon k=1,ldots m}={n_1,n_2}$, $n_1,n_2geq 2$ and $n_1 eq n_2$. In this case, we will "almost'' determine all the families having equations with non-trivial limit cycles. This "almost'' is due to a special family in which we have not been able to determine whether there exist or not non-trivial limit cycles, though we suspect the existence of non-trivial limit cycles.