Planar Quadratic Differential Systems with Invariants of the Form \(a x^2+b x y+c y^2+d x+e y+c_1 t\)

A function I(x, y, t) constant on the solutions of a differential system in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ is called an invariant. We classify all planar quadratic differential systems having invariants of the form \(I(x,y,t)=a x^2+b x y+c y^2+d x+e y+c_1 t\) $I(x,y,t)=a{x}^2+bxy+c{y}^2+dx+ey+c_{1}t$ with \(c_1\ne 0\) $c_1\ne 0$ . There are 13 different families of quadratic systems having invariants of this form. As far as we know this is the first time that quadratic differential systems having an invariant different from a Darboux invariant have been classified