  By a conformal string in Euclidean space is meant a closed critical curve
with non-constant conformal curvatures of the conformal arclength functional.
We prove that (1) the set of conformal classes of conformal strings is in 1-1
correspondence with the rational points of the complex domain $\{q\in
\mathbb{C} \,:\, 1/2 < \mathrm{Re}\, q < 1/\sqrt{2},\,\, \mathrm{Im}\, q >
0,\,\, |q| < 1/\sqrt{2}\}$ and (2) any conformal class has a model conformal
string, called symmetrical configuration, which is determined by three
phenomenological invariants: the order of its symmetry group and its linking
numbers with the two conformal circles representing the rotational axes of the
symmetry group. This amounts to the quantization of closed trajectories of the
contact dynamical system associated to the conformal arclength functional via
Griffiths' formalism of the calculus of variations.
