  In this paper, for a soliton surface $\Omega=\Omega(u,v)$ associated with the
Betchov-Da Rios equation, we obtain the derivative formulas of an extended
Darboux frame field of a unit speed curve $u$-parameter curve
$\Omega=\Omega(u,v)$ for all $v$. Also, we get the geometric invariants $k$ and
$h$ of the soliton surface $\Omega=\Omega(u,v)$ and we obtain the Gaussian
curvature, mean curvature vector and Gaussian torsion of $\Omega$. We give some
important geometric characterizations such as flatness, minimality and
semi-umbilicaly with the aid of these invariants. Additionally, we study the
curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal
(superconformal) Betchov-Da Rios soliton surface with respect to an extended
Darboux frame field. Finally, we construct an application for the Betchov-Da
Rios soliton surface with the aid of an extended Darboux frame field.
