It has been known for a long time that there are close connections between integrable PDEs and CFT. For example, the deep-water-wave equation called the Benjamin-Ono (BO) equation is integrable and its first quantisation sits in a certain CFT. This connection can be understood by the fact that the BO equation can be obtained as a continuum limit of Calogero-Sutherland (CS) models, the eigenfunctions of which also play a role in the diagonalisation of the CFT. Recently a spin generalisation of the Benjamin-Ono equation was proposed and named the half-wave-maps equation. I will discuss how this equation is related to CS models as well as integrable spin chains, thereby further entangling all these different systems and opening the door for investigations into more complex CFTs. I will show what are the most natural generalisations of this equation, indicating the central role played by chirality.