The principal aim is to apply a recently developed fully spectral numerical integrator in order to solve the constraints of the Einstein equations, in their evolutionary forms [1]. This was done by applying the Newman-Penrose “eth”-operator and a multipole expansion of the basic variables in terms of spin-weighted spherical harmonics for the algebraic-hyperbolic and parabolic-hyperbolic forms of the constraints (which in their original form these are partial differential equations) can be traced back to a set of non-linear ordinary differential equations (ODE) for the multipole expansion coefficients [2]. During the term of the PhD program, we intended to develop further our numerical solver to investigate gravitational radiation emitted by dynamical black hole configurations. It is important to be emphasized that half of the research task are fully analytic so candidates with wide enough background have preferences. [1] I.Rácz, Constraints as evolutionary systems, Class. Quantum Gravity 33 015014 (2016) [2] I.Rácz and J. Winicour, Toward computing gravitational initial data without elliptics solvers, Class. Quantum Grav. 35 135002 (2018)