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This is a natural continuation of our first paper \cite{pre}, where wedevelop a new geometrical technique which allow us to study relative equilibriaon the two sphere. We consider a system of three positive masses on$\mathbb{S}^2$ moving under the influence of an generic attractive potentialwhich only depends on the mutual distances among the masses. We reduce theproblem of finding extended Lagrangian relative equilibria to the analysis ofthe inertia tensor, then we obtain a more manageable equivalent inertia tensorwhich allow us to find new families of Lagrangian configurations.

Three-body relative equilibria on $S^2$ II: Extended Lagrangian configurations