DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2021157 DYNAMICAL SYSTEMS SERIES B CORRIGENDUM: ON THE ABEL DIFFERENTIAL EQUATIONS OF THIRD KIND Regilene Oliveira * Departamento de Matem´ atica, ICMC-Universidade de S˜ ao Paulo Avenida Trabalhador S˜ao-carlense, 400-13566-590, S˜ ao Carlos, SP, Brazil Cl´ audia Valls Departamento de Matem´atica, Instituto Superior T´ ecnico Universidade T´ ecnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal (Communicated by Miguel Sanjuan) Abstract. In this paper, using the Poincar´ e compactification technique we classify the topological phase portraits of a special kind of quadratic differ- ential system, the Abel quadratic equations of third kind. In [1] where such investigation was presented for the first time some phase portraits were not correct and some were missing. Here we provide the complete list of non equivalent phase portraits that the Abel quadratic equations of third kind can exhibit and the bifurcation diagram of a 3-parametric subfamily of it. 1. Statement of the main results. For more details about the Abel quadratic differential polynomial of third kind see [1]. The Abel quadratic differential poly- nomial systems of third kind are of the form ˙ x = y 2 , ˙ y = a 0 + a 1 x + a 2 x 2 +(b 0 + b 1 x)y, (1) where a 0 ,a 1 ,a 2 ,b 0 ,b 1 ,b 2 are real parameters with a 2 0 + a 2 1 + a 2 2 = 0. The main result of the paper is the following one. Theorem 1.1. The Abel quadratic polynomial differential equations (1) after a linear change of variables and a rescaling of its independent variable can be written as one of the following systems ˙ x = y 2 , ˙ y = k 0 + k 1 y + x 2 + k 2 xy k 0 ,k 2 ∈ R and k 1 ∈{0, 1}, (i) ˙ x = y 2 , ˙ y = x + k 1 y + k 2 xy k 1 ∈ R and k 2 ∈ {−1, 0, 1}, (ii) ˙ x = y 2 , ˙ y =1+ k 2 xy k 2 ∈ {−1, 1}, (iii) ˙ x = y 2 , ˙ y =1+ k 1 y k 1 ∈{0, 1}. (iv) Moreover, 2020 Mathematics Subject Classification. Primary: 37C15, 37C10. Key words and phrases. Abel polynomial differential equations, vector fields, phase portraits, polynomial solutions. * Corresponding author: Regilene Oliveira. 1