  The parabolic-hyperbolic form of the constraints is integrated numerically.
The applied numerical stencil is $4^{th}$ order accurate (in the spatial
directions) while 'time'-integration is made by using the method of lines with
a $4^{th}$ order accurate Runge-Kutta scheme. The proper implementation of the
applied numerical method is verified by convergence tests and monitoring the
relative and absolute errors determined by comparing numerical and analytically
known solutions of the constraints involving boosted and spinning vacuum single
black hole configurations. The main part of our investigations is, however,
centered on construction of initial data for distorted black holes which, in
certain cases, have non-negligible gravitational wave content. Remarkably the
applied new method is unprecedented in that it allows to construct initial data
for highly boosted and spinning black holes, essentially for the full physical
allowed ranges of these parameters. In addition, the use of the evolutionary
form of the constraints is free from applying any sort of boundary conditions
in the strong field regime.
