= Exact single-phase incompressible perturbation of a shear flow
For a perturbation $\mathbf u'=\tilde{\mathbf u}f(\mathbf k\cdot\mathbf x)e^{st}$ with constant $\tilde{\mathbf u},\mathbf k$ and nonconstant differentiable $f$, <incompressibility> forces $\mathbf k\cdot\tilde{\mathbf u}=0$. Consequently $\mathbf u'\cdot\nabla\mathbf u'=ff'e^{2st}(\mathbf k\cdot\tilde{\mathbf u})\tilde{\mathbf u}=0$. About a shear depending only on $x,z$ and directed along $y$, an axisymmetric such perturbation therefore obeys its linear amplitude equations exactly, including at finite amplitude in this homogeneous model. For <pressure> $P'=\tilde P g(\xi)e^{st}$, nonzero <pressure> requires $g'=Cf$, and $C$ can be absorbed into $\tilde P$; choose $g'=f$. This relation does not imply $g=f$ for an arbitrary profile.
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