  Two-dimensional 3-vector (\textit{d}=2, \textit{n}=3) lattice model with
inversion site symmetry and fundamental group of its order-parameter space
$\Pi_1 (\mathcal{R})= Z_{2}$, did not exhibit the expected topological
transition despite stable defects associated with its uniaxial orientational
order. This model is investigated specifically requiring the medium to host
distinct classes of defects associated with the three ordering directions,
facilitating their simultaneous interactions. The necessary non-Abelian
isotropy subgroup of $\mathcal{R}$ is realized by assigning $D_{2}$ site
symmetry, resulting in $\Pi_1 (\mathcal{R})= \mathbb{Q }$ (the group of
quaternions). With liquid crystals serving as prototype model, a general
biquadratic Hamiltonian is chosen to incorporate equally attractive
interactions among the three local directors resulting in an orientational
order with the desired topology. A Monte Carlo investigation based on the
density of states shows that this model exhibits a transition, simultaneously
mediated by the three distinct defects with topological charge $1/2$
(disclinations), to a low-temperature critical state characterized by a line of
critical points with quasi-long range order of its directors, their power-law
exponents vanishing as temperature tends to zero. It is argued that with
\textit{n}=3, simultaneous participation of all spin degrees through their
homotopically inequivalent defects is necessary to mediate a transition in the
two-dimensional system to a topologically ordered state.
