  The aim of this note is to discuss in more detail the Pohozaev-type
identities that have been recently obtained by the author, Paul Laurain and
Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$
or on the sphere $S^1$ with values into a closed manifold $N^n\subset R^m$.
Weak half-harmonic maps are critical points of the following nonlocal energy
$$\int_{R}|(-\Delta)^{1/4}u|^2 dx~~\mbox{or}~~\int_{S^1}|(-\Delta)^{1/4}u|^2\
d\theta.$$
  If $u$ is a sufficiently smooth critical point of the above energy then it
satisfies the following equation of stationarity $$\frac{du}{dx}\cdot
(-\Delta)^{1/2} u=0~~\mbox{a.e in $R$}~~\mbox{or}~~\frac{\partial u}{\partial
\theta}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $S^1$.}$$
  By using the invariance of the equation of stationarity in $S^1$ with respect
to the trace of the M\"obius transformations of the $2$ dimensional disk we
derive a countable family of relations involving the Fourier coefficients of
weak half-harmonic maps $u\colon S^1\to N^n.$ In the same spirit we also
provide as many Pohozaev-type identities in $2$-D for stationary harmonic maps
as conformal vector fields in $R^2$ generated by holomorphic functions.
