  In numerical relativity simulations with non-trivial matter configurations,
one must solve the Hamiltonian and momentum constraints of the ADM formulation
for the metric variables in the initial data. We introduce a new scheme based
on the standard Conformal Transverse-Traceless (CTT) decomposition, in which
instead of solving the Hamiltonian constraint as a 2nd order elliptic equation
for a choice of mean curvature $K$, we solve an algebraic equation for $K$ for
a choice of conformal factor. By doing so, we evade the existence and
uniqueness problem of solutions of the Hamiltonian constraint without using the
usual conformal rescaling of the source terms. This is particularly important
when the sources are fundamental fields, as reconstructing the fields'
configurations from the rescaled quantities is potentially problematic. Using
an iterative multigrid solver, we show that this method provides rapid
convergent solutions for several initial conditions that have not yet been
studied in numerical relativity; namely (i) periodic inhomogeneous spacetimes
with large random Gaussian scalar field perturbations and (ii) asymptotically
flat black hole spacetimes with rotating scalar clouds.
