We investigate an extension of Harada gravity, a third-order theory proposed as an alternative to General Relativity that preserves energy-momentum conservation and in which the cosmological constant arises as an integration constant. Our aim is to determine whether Harada gravity is unique under the principles that motivated its construction. We show that imposing both the existence of all Einstein solutions and energy-momentum conservation restricts the theory to only two distinct branches: General Relativity and Harada gravity. We then relax each of these assumptions inde pendently and demonstrate that, within the class of symmetric theories considered, Harada gravity remains the unique extension of General Relativity.