I have a complicated 3rd-order ODE of the form $P(y, y', y'', y''') = 0$, where $P$ is a complicated polynomial (5th-order with 24 terms) and coefficients that are (unknown) functions of a parameter $\lambda$, say $$ P(z_0, z_1, z_2, z_3) = \sum c_{j_0 j_1 j_2 j_3}(\lambda) z_0^{j_0} z_1^{j_1} z_2^{j_2} z_3^{j_3}. $$ What I want to know is: Suppose that there exists a non-constant solution $y(x)$ that is independent of $\lambda$. What conditions does this force the coefficient functions $c_{j_0 j_1 j_2 j_3}(\lambda)$ to satisfy?

(An easy analog would be something like: a 1st-order ODE of the form $$y' + c_0(\lambda) y = 0 $$ has nonconstant parameter-independent solutions if and only if $c_0(\lambda)$ is a constant function.)

I'm sure that the specific conditions I'm looking for depend on the precise form of the ODE, which is pretty daunting in this case. I'm just wondering if there's a reasonable algorithm I could apply to find them.