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 \(\dot{x}=a+yz,\quad\dot{y}=-y + x^{2},\quad\dot{z}=b-4x.\) $$\dot{x}=a+yz,\dot{y}=-y+x^2,\dot{z}=b-4x.$$ 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.