The motion of a quantum vortex is considered in the context of the localized induction approximation (LIA). In this context, the instantaneous vortex velocity is proportional to the local curvature and is parallel to the vector which is a linear combination of the local binormal and the principal normal to the vortex line. This implies that the quantum vortex shrinks, which is in contrast to the classical vortex in an ideal fluid. The present work deals with a four-parameter class of static solutions of the equations governing the curvature and the torsion. Such solutions describe vortex lines, the motion of which is equivalent to an isometric transformation. In a particular case when the transformation is a pure translation, the analytic solutions for the curvature and the torsion are found. In the general case, when the transformation is a superposition of a non-trivial translation and rotation, the asymptotics of solutions is explicitly related to the parameters characterizing the transformation, and then to the initial conditions at the zero point of the vortex. In this case, the equations are solved numerically and the shape of a number of different vortices is reconstructed by numerical integration of Frenet–Seret equations.