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In this paper, we give an algebraic construction of a new infinite family ofCameron-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 $q$ isan even power of $3$, we construct the first infinite family of affinetwo-intersection sets in $\mathrm{AG}(2,q)$.