Cameron-Liebler line classes with parameter x=frac{q²-1}{2}

In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$ for $q\equiv 5$ or $9\pmod{12}$, which generalizes the examples found by Rodgers in \cite{rodgers} through a computer search. Furthermore, in the case where $q$ is an even power of $3$, we construct the first infinite family of affine two-intersection sets in $\mathrm{AG}(2,q)$.