  We show the existence of a Lipschitz viscosity solution $u$ in $\Omega$ to a
system of fully nonlinear equations involving Pucci-type operators. We study
the regularity of the interface $\partial \{ u> 0 \}\cap\Om$ and we show that
the viscosity inequalities of the system imply, in the weak sense, the free
boundary condition $u^{+}_{\nu_{+}} = u^{-}_{\nu_{-}}$, and hence $u$ is a
solution to a two-phase free boundary problem. We show that we can apply the
classical method of sup-convolutions developed by the first author in
\cite{caffarelli_harnack_1987,caffarelli_harnack_1989}, and generalized by Wang
\cite{wang_regularity_2000,wang_regularity_2002} and Feldman \cite{Fel} to
fully nonlinear operators, to conclude that the regular points in $\partial \{
u> 0 \}\cap\Om$ form an open set of class $C^{1,\alpha}$. A novelty in our
problem is that we have different operators, $\puccip$ and $\puccin$, on each
side of the free boundary. In the particular case when these operators are the
Pucci's extremal operators $\ppuccip$ and $\ppuccin$, our results provide an
alternative approach to obtain the stationary limit %proof of existence to the
one obtained from of a segregation model of populations with nonlinear
diffusion in \cite{quitalo_free_2013}.
