  We show that a measurable function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, with
$d\geq 3$, satisfies the functional relation \begin{equation*}
g(\omega)+g(\omega_*)=g(\omega')+g(\omega_*'), \end{equation*} for all
admissible $\omega,\omega_*,\omega',\omega_*'\in\mathbb{S}^{d-1}$ in the sense
that \begin{equation*} \omega+\omega_*=\omega'+\omega_*', \end{equation*} if
and only if it can be written as \begin{equation*} g(\omega)=A+B\cdot\omega,
\end{equation*} for some constants $A\in \mathbb{R}$ and $B\in\mathbb{R}^d$.
  Such functions form a family of quantized collision invariants which play a
fundamental role in the study of hydrodynamic regimes of the
Boltzmann--Fermi--Dirac equation near Fermionic condensates, i.e., at low
temperatures. In particular, they characterize the elastic collisional dynamics
of Fermions near a statistical equilibrium where quantum effects are
predominant.
