most general classical theory of gravity

Constructing a field equation for classical gravity based on isotropy and parity invariance suggests the Lagrange-density \(\mathcal{L} = \frac{(\nabla\Phi)^2}{2}+4\pi G\rho\Phi + \lambda\Phi + \sum_n \frac{\Phi^{2n}}{(2n)!}\) What additional (symmetry) principle(s) is required to get the classical Poisson equation?