New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

This study focuses on the integrability and qualitative behaviors of a quadratic differential system ẋ=a+yz,̇=-y + x²,̇=b-4x. x˙=a+yz,y˙=-y+x2,z˙=b-4x. We provide some new perspectives on the system and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.