= Solution
Writing the <uniaxial conductivity tensor> in vector form gives the <electric current density>
$$
\mathbf J=\alpha\mathbf E+\gamma n(n\cdot\mathbf E).
$$
Choose the plane through the origin perpendicular to $n$:
$$
\boxed{\Pi=\{\mathbf E:n\cdot\mathbf E=0\}}.
$$
For an <electric field> in this plane, the anisotropic term vanishes and $\mathbf J=\alpha\mathbf E$. Thus every nonzero such field is parallel or antiparallel to its current according to the sign of the nonzero $\alpha$. The zero field gives zero current. This establishes collinearity throughout the plane without assuming that the tensor is isotropic.
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