In Desktop Integration responsibility, we cannot select from the entire list of Integrators

We are trying to create a new template for the Fixed Asset Additions integrator.  We have assigned the Desktop Integration Responsibility to Fixed Asset user, and they cannot select the integrator needed.

Desktop Integration->Define Layout

In the Select Integrator page, click in the Integrator field and only a very limited selection is available.  We have updated the profile option: BNE: Allow No Security Rule to "Yes".  This increased te number we were able to select, but it still did not include any of the Assets Integrators.

Other users can select from the entire list of Integrators.  The users that can select from the entire also have System Administrator, and Application Developer responsibilities.  We would not be able to give the Fixed Assets users either of these responsibilities.