registering value is not selected in modify mode

Reset mode should be used, but what you're probably experiencing is a double condition. One condition is hiding the field, which resets the value, then another is immediately showing it. Review your conditions carefully as a field should not be conditioned more than once (whether it's conditional others, conditional self, or a tab condition). Also ensure you've latest CB Conditional nightly installed.

your advice is right, but there is a problem, if I set condition only to the tab and not to the field contained within it, in the search form appears the field even if it should appear only as a result of a choice of another field. How can I do to avoid it keeping active reset mode?

You can only condition a field once. It doesn't matter what the condition source is. Whether conditional self, conditional others, or from a tab condition. Review your usage carefully and decide how you want to condition based off this limitation. Tab conditions will typically override any field conditions PHP side.

yes, now I condition a field once. I set a condition only on the tab that contains the field, but now the problem is search form (see the attached file)

Is it possible integrate CB conditional and cb core fields ajax plugin, so if I modify a field from frontend with core fields ajax plugin, reset mode start for conditional fields?

yes, now I condition a field once. I set a condition only on the tab that contains the field, but now the problem is search form (see the attached file)

You can only condition a field once. How you condition it knowing this limitation is entirely up to you. If you need field by field conditioning then I recommend applying the conditions to the fields directly instead of the entire tab.

Is it possible integrate CB conditional and cb core fields ajax plugin, so if I modify a field from frontend with core fields ajax plugin, reset mode start for conditional fields?

Nope. I've looked into this for a future release and support for it won't be implemented. It would require a complete rewrite to how CB Conditional bindings work and that's just not going to happen. Ajax usages will take a page reload for the condition to be visible. In addition to this for profile view fields already hidden are completely removed from the fields output so even if a condition made it show it wouldn't exist in the DOM to show.