  We consider the problem of existence and uniqueness of strong a.e. solutions
$u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE
system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on
}\mathbb{R}^n, \] when $ f\in L^2(\mathbb{R}^n)^N$ and $F$ is a Carath\'eodory
map. \eqref{1} has not been considered before. The case of bounded domains has
been studied by several authors, firstly by Campanato and under Campanato's
ellipticity condition on $F$. By introducing a new much weaker notion of
ellipticity, we prove solvability of \eqref{1} in a tailored Sobolev "energy"
space and a uniqueness estimate. The proof is based on the solvability of the
linearised problem by Fourier transform methods, together with a "perturbation
device" which allows to use Campanato's near operators. We also discuss our
hypothesis via counterexamples and give a stability theorem of strong global
solutions for systems of the form \eqref{1}.
