Simultaneity of centres in double-reversible planar differential systems

Abstract

ABSTRACT We provide the sufficient conditions for the simultaneous existence of centres for two families of planar double-reversible quintic systems. Computing the focal values and using crossed resultants and Gröbner bases, we find the centre conditions for such systems. The results obtained show that there exist a kind of symmetry where the number of centers in a quintic system is more than 5, so higher than the degree of the system. This fact gives a counterexample to the question posed in a previous work.