In this paper we study the existence and uniqueness of limit cycles for socalled quadratic systems with a symmetrical solution: dx(t) dt = P2(x, y) ≡ a00 + a10x + a01y + a20x 2 + a11xy + a02y 2 dy(t) dt = Q2(x, y) ≡ b00 + b10x + b01y + b20x 2 + b11xy + b02y 2 where (x, y) ∈ R2 , t ∈ R, aij, bij ∈ R, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic.
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