  In this work, we study the existence of $C^{\infty}$ local solutions to
$2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right
hand side function $f$ possibly vanishes, changes the sign, is positively or
negatively defined. We also give the convexities of solutions which are related
with the annulation or the sign of right-hand side function $f$. The associated
linearized operator are uniformly elliptic.
