  We study the distribution of orbits of a lattice
$\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolume
one rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and our
main result is the limiting distribution of these orbits with respect to norm
balls, where the norm is given by the sum of squares. Specifically, we consider
$\Gamma_T=\{\gamma\in\Gamma:\|\gamma\|\leq T\}$ and show that, for any fixed
$x_0\in X_{2,3}$ and $\varphi\in C_c(X_{2,3})$,
$$\lim_{T\to\infty}\frac{1}{\#\Gamma_T}\sum_{\gamma\in\Gamma_T}\varphi(x_0\cdot\gamma)=\int_{X_{2,3}}\varphi(x)d
\tilde\nu_{x_0}(x),$$ where $\tilde\nu_{x_0}$ is an explicit probability
measure on $X_{2,3}$ depending on $x_0$. To prove our result, we use the
duality principle developed by Gorodnik and Weiss which recasts the above
problem into the problem of computation of certain volume estimates of growing
skewed balls in $H$ and proving ergodic theorems of the left action of the
skewed balls on $\text{SL}(3,\mathbb{R})/\Gamma$. The ergodic theorems are
proven by applying theorems of Shah building on the linearisation technique.
The main contribution of the paper is the application of the duality principle
in the case where $H$ has infinitely many non-compact connected components.
