The following code does what you seem to need (screenshots attached). The following modules need to be activated:

 - i18n_menu (part of the i18n)

Further configuration: 

 - I added a new language (tr) and set language detection to "Determine the language from the URL (Path prefix or domain)."
 - I added a new menu with machine name (menu-sol-menu) and set the translation mode to "Translate and Localize". Also added the new menu as a block to the header region of the theme.
 - Added the following code in a custom module and activated it.


        function agi_menu() {
          $items = [];
          $items['agi'] = [
            'title' => 'AGI (Family Status Declaration)',
            'page callback' => 'drupal_get_form',
            'page arguments' => ['node_form'],
            'access arguments' => ['access content'],
            'menu_name' => 'menu-sol-menu'
          ];
          return $items;
        }



I visited the menu admin page and saw that a new menu item was added:

[![enter image description here][1]][1]

I checked the language option for that new menu item, it was set to neutral.

[![enter image description here][2]][2]

I visited the home page of my drupal instance and checked that the menu item appears, for both english (default language) and turkish, using the URL to define the current language.

[![enter image description here][3]][3]


Finally, I looked in the translate interface for the title of the menu item.

[![enter image description here][4]][4]


Conclusion: The provided code with the described configuration works as expected by the OP.


Good luck! 


N.B.: Especially for the 'title' and 'title callback' properties, the [documentation][5] is very clear about how to use them in a multilingual setting

> "title": Required. The untranslated title of the menu item. "title
> callback": Function to generate the title; defaults to t(). If you
> require only the raw string to be output, set this to FALSE.


  [1]: https://i.sstatic.net/PZkPs.png
  [2]: https://i.sstatic.net/ynwVy.png
  [3]: https://i.sstatic.net/26ex8.png
  [4]: https://i.sstatic.net/MZBaG.png
  [5]: https://api.drupal.org/api/drupal/modules%21system%21system.api.php/function/hook_menu/7.x