Getting rid of first degree to apply for second
Don't throw out something you already have. If your goal is to do research and write some papers, then you can in essence do that in your spare time. If you're good, people will want to collaborate with you and mentor you whether you're a graduate student or not. In other words, I don't see the reason why you want to go through graduate school a second time -- having to take all of the classes again, take qualifying exams again, etc. All of that doesn't help you in your goal, for which you are already qualified because of your previous PhD.
Of course, it may be that you want to go back to be a grad student because it usually also includes a stipend, health insurance, etc. But you can get that from other sources as well: You could do research in your spare time, or simply have a part-time job so you have more time for research. A good part-time job may in fact pay no less than a grad student salary.
A particularly useful choice for a part-time position would, of course, be within a math department itself. For example, you could seek to be a lecturer in a math department, where you teach let's say half of the usual load, for half the salary, leaving you the other half of your time for research and paper writing. This would also give you access to professors with whom you may want to work, and you would likely be able to audit courses for free if you're already a member of the department.
Can I somehow revoke or renounce my degree?
No, and no. By which I mean:
1) There is no honorable way to formally divest yourself of an academic degree.
The only way that academic degrees get revoked is through misconduct, usually academic misconduct. (As has come up on this site before, in theory an institution which discovers that a former student has not paid all their fees might try to revoke the degree. But I have never seen this happen, and anyway PhD students in mathematics rather usually receive money from their university.) Obviously this is not part of any sound plan to regain entry into academia.
2) Even if you somehow could get your degree "annulled," that would not have the intended effect: you would still (in my opinion, obviously) be obligated to divulge to all PhD programs your prior schooling, send them your transcripts, and so forth. Giving them all this information and then saying "But then I got my degree annulled" is not going to satisfy anyone. What you seem to be contemplating -- I think; it's the only thing that could work for you if it were kosher -- is to make some kind of formal arrangement between you and your PhD-granting institution that both parties can act as if the PhD there never happened. But this is basically an agreement with one party to lie (or mislead, at least) about your past to a bunch of other parties: in other words, it's inherently ethically suspect in a manner that could really push the buttons of the people who would be evaluating you.
Maybe it helps to think of it this way: when I say "I have a PhD in mathematics," I am not describing a present rank or title. I am calling attention to a past event. The past is, well, passed, so if you got a PhD at any point then you still have it. (Even if your PhD gets revoked for academic misconduct, I would like to say that you still have a PhD -- e.g. you still have a paper document that says that -- it's just been decided that you are not entitled to any of the favorable consequences of that degree.) To try an analogy: if I ask my date "Have you ever been married?" and she got married once and had the marriage annulled, the honest answers are "Yes." and "Yes, but it was annulled." To say "No" is deceptive: I am asking about her past, not her current legal status.
I think the broader questions here are "Why do you want to go into academia?" and "What exactly is it, within academia, that you want to do?" Are you interested in research, or teaching? Does the field matter to you? Both questions are potentially hugely important here.
For instance, if you want to go into academia to teach introductory math classes, then perhaps you should just try to get a job teaching in a local community college or high school.
On the other hand, if what you want is a career as a researcher, then you really need to decide which field you're passionate about--pure math, or something else. If it's pure math you love, then forget doing a second doctorate---just spend that time publishing and attending conferences. If it's something else that you love, perhaps you could craft a compelling rationale for why you need a second degree in order to pursue your new interest. I've seen that happen sometimes, but you need to have a very specific rationale for why you need this new degree in a new field to pull it off.
I worry though, from the tone of your post, that what you're really in love with is just some kind of abstract idea that academia > industry and that you think you'd really love the life of a career researcher. I think that's dangerous, because for the vast majority of full time faculty who make up academia, teaching and service obligations are much larger parts of the job than research. So, my advice is to first sit and think really hard about what exactly you want to do and why. Once you know the answers to those questions, your path will become much clearer.