A Classification of the flag-transitive $2$-$(v,3,\lambda )$ designs with with $v\equiv 1,3(\mathrm{m}\mathrm{o}\mathrm{d}6)$ and $v\equiv 6(\mathrm{m}\mathrm{o}\mathrm{d}\lambda )$

In this paper, we provide a complete classification of the $2$-$(v,3,\lambda )$ designs with $v\equiv 1,3(\mathrm{m}\mathrm{o}\mathrm{d}6)$ and $$ admitting a flag-transitive automorphism group non-isomorphic to a subgroup of $A\mathrm{\Gamma }{L}_1(v)$.