The initial data of the vacuum Einstein equations on a spacelike hypersurface satisfy the Einstein constraint equations. It is an underdetermined PDE system. We exploit the underdeterminedness to find nontrivial solutions of the Einstein constraint equations with interesting properties. A key ingredient is the construction of Bogovskii-type operators for underdetermined PDEs satisfying a robust symbol condition. The Bogovskii-type operators have favorable support and regularizing properties, allowing us to manipulate the solutions flexibly. This is based on joint work (partly in progress) with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Sung-Jin Oh (UC Berkeley).