  We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope
$\alpha'$. Actually we obtain at the same time a solution of the open case
$\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions
for a family of Hessian equations which includes the Fu-Yau equation as a
special case. The method is based on the introduction of a more stringent
ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and
which can be shown to be preserved by precise estimates with scale.
