Orbit of a one-parameter proper Euclidean motion

The integral curves of $V(x)=a\times x+b$ are lines when $a=0$, and helices around an axis when $a\ne 0$. The helix is closed exactly when its axial translation component vanishes, in which case every nonfixed orbit is a circle.

A simple closed curve in ${\mathbb{R}}^3$ preserved by a nonconstant smooth family of proper rigid motions is a circle. Differentiating the normalized family at the identity gives a nonzero infinitesimal proper Euclidean motion tangent to the curve. Compactness rules out its line and nonclosed helical orbits, leaving a circular rotation orbit.