in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by,

\begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{align}

Here $$v$$ is the constant speed and $$\kappa$$ is the curvature. I don't see any problem with these equations if $$\kappa = 0$$, but I have read that the frame is not defined if curvature is zero.

As long as the curve is regular (has nonzero velocity at each point) you can always define a right-handed $$T,N$$ frame everywhere. But you must allow curvature to change sign. (Ordinarily, we always define $$\kappa\ge 0$$.)