= Solution
The condition $\mathbf E_I\cdot\widehat{\mathbf x}=0$, together with <transversality of an electromagnetic plane wave>[transversality] $\mathbf k_I\cdot\mathbf E_I=0$, selects polarization along $\widehat{\mathbf y}$ away from the normal-incidence degeneracy. <Phase matching at a planar wave interface> gives
$$
\omega_T=\omega_I,
\qquad k_{Ty}=0,
\qquad k_{Tz}=k_{Iz}=k_I\sin\theta_I.
$$
The two <dispersion relations> give $|\mathbf k_T|^2=(\varepsilon_+/\varepsilon_-)k_I^2$. Choosing propagation into $x>0$ therefore yields
$$
\boxed{
\mathbf k_T=k_I\left[
\sqrt{\frac{\varepsilon_+}{\varepsilon_-}-\sin^2\theta_I}\,\widehat{\mathbf x}
+\sin\theta_I\,\widehat{\mathbf z}
\right]}.
$$
Equivalently, this is <Snell law for electromagnetic waves> with <refractive index> proportional to $\sqrt\varepsilon$ because the two media have equal <magnetic permeability>.
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