Newton invariants

one can in principle construct infinitely many invariants of the tidal shear \(\Phi_{ij}\equiv \partial^2_{ij}\Phi\) by considering traces of powers of the tidal shear matrix, \(\mathrm{tr}(\Phi^n)\), where one recovers the Poisson equation \(\Delta\Phi = 4\pi G\rho\) for \(n=1\). are there physical interpretations for other values of \(n\)?