Good morning everybody. My question is inspired from the following fact:

Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial_x^2 g(x,y,z)=0$ we would find $g(x,y,z)=x p_0(y,z)+p_1(y,z)$. Then it will be possible, playing with $p_0$ and $p_1$, to prescribe for example both the behaviour on the hyperplanes $x=0$ and $x=1$ (Am I right?)

The question then is as follows: Suppose $X,Y,Z$ to be three arbitrary linearly independent vector fields (still in $\mathbb R^3$ with standard coordinates) and assume we want to solve the equation $X^2 g(x,y,z)=0$.

Assume $\pi_0$ and $\pi_1$ to be two hypersurfaces transverse to $X$ (they play the same roles of the previous hyperplanes).

Is it possible as above to find $g$ that satisfies ALSO prescribed conditions both on $\pi_0$ and $\pi_1$?

Intuitively this seems to be right to me, however I am not in possess of a rigorous proof, nor I know enough general theory to convince myself (therefore references are welcomed).

Thanks in advance for your kindness.

Regards,

-Guido-

Edit In response to prof. Bryant comment, let's say that my case of interest concerns with $X=\partial_x-\frac y2\partial_z$, but still I have no particular constraints on $\pi_0$ and $\pi_1$, except for them being different and both transverse to $X$.