Equations of $G$-minimal conic bundles

We develop a method for obtaining equations for $G$-minimal conic bundles with an arbitrary number of singular fibres. When the number of singular fibres is equal to $4$, $6$, or $7$, a detailed classification is given, which includes obtaining the equations for minimal conic bundles $(S,G)$ and an explicit indication of the action of the group $G$ on the Picard group $\operatorname{Pic}(S)$ and on the surface $S$ itself.
Bibliography: 19 titles.