There are no exact solutions of Einstein Equations that describes a bound system radiating gravitational waves. One needs to resort to numerical simulations, or analytical approximation methods. Current methods to constrained initial data exhibit junk radiation and ambiguities about constrained and free data. It was mathematically proved that given the correct initial data, Einstein equation will yield the expected solution.
Hamilton, M. & Winicour, J. (2017, June). New Numerical Code for Black Hole Initial Data. Slides presented at the 20th Eastern Gravity Meeting. Penn State, PA.