In the present paper, we study the reversible limit cycles for the system \(\begin{aligned} \begin{array}{ccl} {\dot{x}}& & = a_1 x+a_2 y + a_3x^3 + a_4x^2y + a_5xy^2+ a_6 y^3,\\ {\dot{y}}& & = b_1x+b_2y +b_3x^3 + b_4x^2y + b_5xy^2 + b_6y^3, \end{array} \end{aligned}\) which is called a linear plus cubic homogeneous polynomial differential system. It is proved by Zhou that the reversible limit cycles of a polynomial differential system are algebraic. We show that the degree of the reversible limit cycle of the above system is at most 6. Moreover, this system has no reversible limit cycles of degree 3 with respect to straight lines passing through the origin in the phase plane. We present a system showing that the above system can have two reversible limit cycles of degree 4.
