Antiplane energy does not determine the full stress

On an antiplane shear $F=I+e_3\otimes (p,q,0)$, the first strain invariant satisfy $I_1=I_2=3+p^2+q^2$, $I_3=1$. Restricting an isotropic objective strain energy density to this family determines only derivatives tangent to the family. It fixes $S_{31}=2w^{\prime }(p^2+q^2)p$ and $S_{32}=2w^{\prime }(p^2+q^2)q$, but does not specify the whole Cauchy stress tensor. For example the neo-Hookean solid energy $W=\mu (|F{|}^2-3)/2-\mu \log \det F$ gives $S=\mu (B-I)$ on this family. Even though the reference state is stress free, this is not generally a scalar multiple of $B$.