Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$

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 $9(\mathrm{m}\mathrm{o}\mathrm{d}12)$, 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 $\mathrm{A}\mathrm{G}(2,q)$.