For some remarks on what the general story looks like you might want to consult the Princeton Companion article about partial differential equations. I don't have much to say myself about this so I'll just mention two examples.
The Korteweg-de Vries equation is of interest to people working in integrable systems (there are other examples at that link). I understand that it has some interesting ties to algebraic geometry.
Another example (really a family of examples) is the following. Let $G$ be a Lie group acting smoothly on a smooth manifold $M$. This induces a map from the Lie algebra $\mathfrak{g}$ to the Lie algebra of smooth vector fields on $M$, which induces an action of the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ on $C^{\infty}(M)$ by differential operators. Choose $D \in U(\mathfrak{g})$. Then you can write down the differential equation $Df = 0$
where $f \in C^{\infty}(M)$. This differential equation should have order the degree of $D$ (with respect to the usual filtration on $U(\mathfrak{g})$). The space of solutions to this differential equation is acted on by the subgroup of $G$ commuting with $D$. In particular, if $D$ lies in the center, the space of solutions is acted on by all of $G$.
For a very simple example, let $G = \mathbb{R}$ act on itself by translation. Then $\mathfrak{g}$ is spanned by ordinary differentiation $D$, the universal enveloping algebra $U(\mathfrak{g})$ is polynomials in $D$, and the corresponding differential equations you can write down are all linear homogeneous ODEs with constant coefficients, all of which admit an action of $G$ (again by translation). Writing down this action explicitly gives generalizations of the angle addition formulas (which you obtain in the special case $D^2 + 1$).
For a non-abelian example, let $G = \text{SO}(3)$ act on $\mathbb{R}^3$ by rotation. Then the image of $\mathfrak{g} = \mathfrak{so}(3)$ in vector fields on $\mathbb{R}^3$ is spanned by
$L_x = y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y}$ $L_y = z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}$ $L_z = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}.$
These vector fields generate rotation about the $x, y, z$-axes respectively. The Casimir operator $L^2 = L_x^2 + L_y^2 + L_z^2$
turns out to generate the center of $U(\mathfrak{so}(3))$, so the space of solutions to the differential equation $p(L^2) f = 0$ for any polynomial $p$ is acted on by all of $\text{SO}(3)$ and has order $2 \deg p$.