In my research I have stumbled across the following 1st order complex differential equation for smooth functions $\eta:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}-\lbrace0\rbrace$ defined on the circle,
$$i\frac{\partial\eta}{\partial t}+(re^{it}+\varepsilon i)\bar\eta=0$$
where $\varepsilon\in\mathbb{R}_+$ is sufficiently small and *fixed* (so choose $0<\varepsilon<<1$ but maybe not $\varepsilon=\frac{1}{2}$) and $r\in\mathbb{R}_+$ is a nonzero positive real *parameter*. Motivated by some perturbation theory and functional analysis, I believe there must exist a nontrivial solution $\eta$ to this ODE for *at least one* choice of $r$. **What is such an explicit pair $(r,\eta)$?**

This problem arises when studying the asymptotics of punctured $J$-holomorphic curves, and I need the solutions to do what I want to do (when perturbing $J$). Ultimately I desire the set of all such distinct $r$ and the dimension of the kernel of the corresponding differential operators.

$\underline{\text{Attempt}}$ I originally posted this on StackExchange a month ago with many edits but not much luck. My attempt was to Fourier expand $\eta(t)=\sum_{k\in\mathbb{Z}}a_ke^{ikt}$ and obtain the recurrence relation $$-ka_k+r\bar a_{1-k}+\varepsilon i\bar a_{-k}=0$$ I need a specific collection $\lbrace a_k\rbrace\subset\mathbb{C}$ which solves this (for some $r>0$). What I get at the least is $r=\varepsilon i\frac{a_0}{a_1}$ (and subsequently $\frac{a_0}{a_1}$ must be purely complex and nonzero). But there is still a good chance that the coefficients $a_k$ will "blow up" as $k\to\infty$ if not chosen carefully. I've done some manipulations and I cannot parse whether they are helpful or harmful.

Also, this *complex* ODE is equivalent to two coupled *real* ODEs, but I don't think it helps. Decompose $\eta=x+iy$ and attempt to solve the equivalent system:
$$\dot y(t) -r\cdot\cos(t)\cdot x(t) -[\varepsilon +r\cdot\sin(t)]\cdot y(t) = 0$$
$$\dot x(t) -r\cdot\cos(t)\cdot y(t) +[\varepsilon +r\cdot\sin(t)]\cdot x(t) = 0$$
Perhaps there are numerical methods to find "approximate" periodic solutions, or some software to plot $(x,y)$ for various values of $r\in\mathbb{R}_+$?

realcoefficients because if you have a complex solution, the real parts of all variables will solve the same system and the imaginary parts will solve a slightly modified system, so the complex equation splits completely into two real ones. BTW, I took one determinant with a wrong sign when computing, so I am no longer sure a solution exists (but still sure that the technique works one way or another). I'll post the details when I have some free time; too busy now... $\endgroup$ – fedja Sep 11 '17 at 20:48