A sufficient and necessary condition for $\mathcal{A}$-quasiaffinity

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.