I will discuss the energy-critical half-wave maps equation. This is a novel geometric evolution equation, which arises as a universal continuum limit for completely integrable spin systems. My talk will focus on two aspects: i) The explicit classification of all traveling solitary waves, and ii) the complete spectral analysis of the associated linearized operator. As for i), we will exploit an intriguing connection to minimal surface theory. The proof of ii) will make use of a conformal transformation that reformulates the spectral problem in terms of Jacobi operators (i.e. infinite tridiagonal matrices) posed on the unit circle.