  We consider a homogeneous, constant rank differential operator $\mathcal{A}$
and prove a characterisation theorem for $\mathcal{A}$-quasiaffine functions in
the spirit of Ball, Currie and Olver (1981); i.e. functions such that \[ f(v) =
\int_{T_N} f(v + \psi(y))~\mathrm{d}y \] for all $v$ and all $\mathcal{A}$-free
test functions $\psi$ with zero mean. This result is used to get a sufficient,
but not necessary condition for the differential operator $\mathcal{A}$, such
that linearity along the characteristic cone of $\mathcal{A}$ implies
$\mathcal{A}$-quasiaffinity. We show that this implication is true if
$\mathcal{A}$ admits a first order potential.
