OK, with the motivation given by the OP in the edited question here is how I would approach the job interview with your would-be advisor. The worst thing you could do is to fake advanced knowledge when you lack basic understanding. I am sure you will make a much better impression if you admit that the paper you were asked to study is far from your expertise, but that you have made an effort to understand the basics you would need to pursue that line of research. So the first step would be to learn what is Darboux integrability. This book chapter could be a good entry point: Darboux integrability, but I'm sure you'll find other entry points if you search the web.
The topic of the paper you were asked to study is to find a way to discretize this special nonlinear differential equation referred to as a Laine equation, without spoiling the existence of the integrals of motion that are the defining property of Darboux integrability. At this point your would-be advisor might ask: "Is the Laine equation Darboux integrable", and it would be great if you could then go to the blackboard and show that it is.
I'm pretty sure that would be sufficient for the interview: you have shown you can place a new topic in its proper context and that you can do the background reading you need in order to enter this new field. If you want to go further, and you have the time, you could see if you can make sense of the discretization procedure advanced in the paper. But that would be optional, I think.