Classification of flag-transitive primitive symmetric $(v,k,\lambda )$ designs with $PSL(2,q)$ as socle

Let $\mathcal{D}$ be a nontrivial symmetric $(v,k,\lambda )$ design, and $G$ be a subgroup of the full automorphism group of $\mathcal{D}$. In this paper we prove that if $G$ acts flag-transitively, point-primitively on $\mathcal{D}$ and $Soc(G)=PSL(2,q)$, then D has parameters $(7,3,1)$, $(7,4,2)$, $(11, 5, 2)$,$(11, 6, 3)$or$(15, 8, 4)$.