On $2$-ordered groups

Let $h(x, y, z)$ denote the standard orientation of the plane $\mathbf{R}^2$. Let $M$ be a non-empty set, $\zeta: M\to\{0, +1, -1\}$.
If for every subset $A$ of a set $M$, $|A|\leqslant 5$, there exists a map $\phi: A\to\mathbf{R}^2$, such that $x, y, z\in A$ implies
$$ \zeta(x, y, z)=\eta(\phi(x), \phi(y), \phi(z)), $$
then $(M, \zeta)$ is called a $2$-ordered set and $\zeta$ is called a $2$-order function on $M$.
If $\zeta$ is a $2$-order function on a group $G$ such that for every $x, y, z, a$ from the group $G$ the equality
$$ \zeta(ax, ay, az)=\zeta(xa, ya, za)=\zeta(x, y, z) $$
holds, then $G$ is said to be a $2$-ordered group.
The paper contains new examples of $2$-ordered groups. It is proved that every $2$-ordered group contains only one involution or none. A criterion is formulated for a straight line in a $2$-ordered group $G$ to be a subgroup of $G$.