So, it dopends on how much your permit means to you... if I were you, I would get a copy of your driving record (according to DMV) along with any documentation you have that backs up your story and then start a time line. On blah blah blah date, my TWIN BROTHER, (fill in name here), received a ticket for ______. He used my name. Basically, start explaining your story incident by incident to help the investigator out. If you can put it into something that is clear, cut and dry (leave out the emotion), call the investigator back and be as helpful as you can in helping him sort things out.
Then again, if you dont want to put the time into all of this, then in my own opinion (which basically means nothing LOL), you'll never get a permit.
Moosie