So the result should obviously show input type="hidden" not engine="hidden" for it to be not printed in the front end. I personally would try renaming the ENGINE to type within the snippet, although that’s just a blind shot and might break things.
Unfortunately I’m not familiar with the AjaxForm Extra (as all the docs are russian) so I can’t really help much but maybe you could post your form call and related templates here, so we can take a closer look.
Thank you for your willingness to help
the fact is that everything worked fine, I just changed the hosting, the code itself did not touch. and this field appeared.