I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new proof of this result? (So far I have found the article by Kneser, the article by Nelson G. Markley and some works that use the results of these articles, I am looking for some other idea focused on proving the result mentioned at the beginning) I would really like to read this result, I tried to do it from your original article but the language is too complicated for me. I searched on the internet but found almost nothing about the proof. I asked here but I didn't find any answer even with bounty. I hope to have some help it would help me a lot.
The answer below was given to the question as asked originally.
For a more modern, english language proof of Kneser's result, see The Poincaré-Bendixson Theorem for the Klein Bottle, by Nelson G. Markley:
In 1923 Kneser showed that a continuous flow on the Klein bottle without fixed points has a periodic orbit. The purpose of this paper is to prove a stronger version of this theorem. It states that the Klein bottle cannot support a continuous flow with recurrent points which are not periodic.
It sounds like you are looking for a textbook. I have read bits of Introduction to the geometry of foliations: Part A by Hector and Hirsch. It is well-written, with pictures! They give Kneser's theorem on pages 62-65, after developing the necessary theory.