  In this paper, we mainly focus on the existence of the viscosity solutions of
\begin{equation*}
  \left\{
  \begin{aligned}
  &H_1(x,Du_1(x),u_1(x),u_2(x))=0,\\
  &H_2(x,Du_2(x),u_2(x),u_1(x))=0.
  \end{aligned}
  \right. \end{equation*} The standard assumption for the above system is
called the monotonicity condition, which requires that $H_i$ is increasing in
$u_i$ and decreasing in $u_j$ for each $i,j\in\{1,2\}$ and $i\neq j$. In this
paper, it is assumed that $H_i$ is either increasing or decreasing in $u_i$,
and may be non-monotone in $u_j$. The existence of viscosity solutions is
proved when \[\chi:=\sup_{u,v,w\in\mathbb R}\bigg|\frac{\partial_{u_2}
H_1(x,0,0,u)}{\partial_{u_1} H_1(x,0,v,w)}\bigg|\cdot \sup_{u,v,w\in\mathbb
R}\bigg|\frac{\partial_{u_1} H_2(x,0,0,u)}{\partial_{u_2}
H_2(x,0,v,w)}\bigg|<1.\] Then we consider \begin{equation*}
  \left\{
  \begin{aligned}
  &h_1(x,Du_1(x))+\Lambda_1(x)(u_1(x)-u_2(x))=c,\\
  &h_2(x,Du_2(x))+\Lambda_2(x)(u_2(x)-u_1(x))=\alpha(c).
  \end{aligned}
  \right. \end{equation*} It turns out that for each $c\in\mathbb R$, there is
a unique constant $\alpha(c)\in\mathbb R$ such that the above system has
viscosity solutions. The function $c\mapsto \alpha(c)$ is non-increasing and
Lipschitz continuous. In the appendix, the large time convergence of the
viscosity solution of evolutionary weakly coupled systems is proved when
$\chi<1$.
