Ah, the dreaded error. Dealing with the city govt will be a pain. You have a notice of showing up (you still have it right? - you will be amazed at how easily documents that prove their mistakes find themselves deleted from their systems) but does it say that you were found not guilty? Could be that the mistake was done when you showed up to court, or even if it had been filed under a different license number by the officer and later corrected by some pencil pusher.
This is just a speeding ticket right? More than likely they'll just tack on points to your license and submit the fine to a collection agency, who will then begin to of course hound you (which is what this: "This action is independent of any other action taken by the court or this department." could mean).
Once it goes to collections, you'll be better off paying it. Collection agencies in this country are crooks and they know it...whether you owe or not, they'll just harass you until you pay knowing full well that the amount is usually far less than it would cost you to defend it. Even when the courts order it off, it sometimes just gets sold to someone else and you have to start over.
Worse part, they can just print up anything they want if they have your information (which of course the DMV does). If you dont have your copy, you are stuck with whatever error they made. I did business with a phone company once, cancelled, paid the last bill and moved out, and 2 years later got a collection for $50 for service when I wasnt even living there. Of course no one cared, and even the phone company said, "well, we had you as cancelling on such and such date, not this other date, whether you lived there or not doesn't matter because you still had the phone in your name"...since they never give you 'cancellation paperwork' and most refuse down to the supervisor not to fax you or send you ANYTHING, you are at their mercy.
The real kicker? A year later I got ANOTHER bill from them for a smaller amount. I just had to pay. It stopped after that, but I'm telling you, the law doesn't work for ordinary folks like us when dealing with huge companies and small bills. I even called lawyers the second time around to ask advice, they told me to pay it to protect my credit because no one on earth would represent me for such a small amount.
Welcome to the real world! You're going to get screwed all the time, best get used to it
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PS: Are you sure its really from the DMV? I got a letter last week from a unnamed address that told me my homestead exemption had not been filed and I needed to send $35 for the privilege. Freaked me out for a minute, and the whole letter seemed to be legit. But at the end in fine print it said "this is independent of any tax homestead exemption you may have filed"...have to scrutinize these things carefully....I dont know how they get away with this, but its legal to trick people I suppose.