= Solution
Let $K^2=|\mathbf k|^2=k_x^2+k_z^2+\Omega^2t^2k_x^2$. For the complex wave before taking its <real part>,
$$
(\partial_t+\Omega y\partial_x)(\mathbf v e^{i\mathbf k\cdot\mathbf x})=\left[\dot{\mathbf v}+i(\dot{\mathbf k}\cdot\mathbf x+\Omega yk_x)\mathbf v\right]e^{i\mathbf k\cdot\mathbf x}=\dot{\mathbf v}e^{i\mathbf k\cdot\mathbf x},
$$
since $\dot k_y=-\Omega k_x$. The asserted <velocity> equation therefore follows directly from $\dot{\mathbf v}+\Omega v_y\widehat{\mathbf x}=-i\mathbf kq$. To verify the <pressure> amplitude, solenoidality is $\mathbf k\cdot\mathbf v=0$. Its derivative requires
$$
0=\dot{\mathbf k}\cdot\mathbf v+\mathbf k\cdot\dot{\mathbf v}=-2\Omega k_xv_y-iK^2q,
$$
and hence $q=2i\Omega k_xv_y/K^2$, as claimed.
The transverse amplitude equations are
$$
\dot v_y=av_y,\qquad\dot v_z=bv_y,\qquad a=\frac{2\Omega k_xk_y}{K^2},\quad b=\frac{2\Omega k_xk_z}{K^2}.
$$
Both coefficients are real. Differentiating $v_yv_z^*-v_zv_y^*$ makes the terms proportional to $b|v_y|^2$ cancel, while the remaining terms give $a(v_yv_z^*-v_zv_y^*)$. Therefore the <helicity invariant of an inviscid shearing wave> obeys
$$
\boxed{\dot H=\frac{2\Omega k_xk_y}{K^2}H=-\frac{2\Omega^2tk_x^2}{k_x^2+k_z^2+\Omega^2t^2k_x^2}H,\qquad H(t)=\frac{H_0(k_x^2+k_z^2)}{k_x^2+k_z^2+\Omega^2t^2k_x^2}.}
$$
Here $H$ is real because the difference in its definition is purely imaginary. Its sign is preserved, and shear reduces its magnitude away from $t=0$.
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