Integrability and non-linearizability of weak saddles in a cubic Kolmogorov model

•The necessary and sufficient integrability conditions is given.•The non-linearizability is characterized.•Integrability conditions of weak saddles are compared with existing integrability condition of weak foci.•To identify the integrability and non-linearizability, computational algebraic geometry techniques such as the Theory of Resultant and Gröbner Bases are used.