Wow, the modern man is really in danger of getting lost in his own abstractions! :) There is no "desire of no desire" in Buddhism, it is something people's logical minds make up out of mathematical relationships between things they hear about. There is no "desiring to have no desire". There is no "striving to abolish desire" either.
Here is my understanding:
What there is, is desire to be free of pain, desire to not be driven by troubles - the desire of peace. Consequently, there is striving for peace. There is nothing contradictory in striving for peace. You need to identify the obstacles to peace and remove them. Accepting the obstacles to peace will not lead to peace.
In Buddhism, the obstacles to peace are, starting from the coarse ones: actions like stealing, violence, lying and verbal abuse, sexually-motivated trouble, and getting intoxicated.
It is pretty clear that these cause trouble, right?
Next, we get to understand that the cause of trouble is not just our behavior, it is our mind, our attitude. So we speak about the Three Root Poisons: attraction, repulsion, and confusion. Or obsession, aversion, and delusion. Or lust, hate, and ignorance. Many partial synonyms, same idea. Our minds are colored by various mixtures of these three "kleshas" - as a result we see things in wrong light - and - get in trouble.
Next, we understand that the above really boils down to getting stuck on stuff, what's known as attachment. This is a very broad category. Getting stuck is when reality moves on, but your mind is still in the old place. Or when, reality does not move, but your mind wants to get to some imaginary point. From this mismatch between "is" and "should" comes "dukkha" the feeling of wrongness, colloquially known as suffering. This is the First and Second Noble Truths, dukkha and origination of dukkha.
All things exist in dependence on some necessary and sufficient conditions. When the necessary conditions are no longer in effect, the things can no longer exist. This gives us leverage over things, including leverage over dukkha. By dropping the attachment we remove the basis of dukkha -- the mismatch, and dukkha disappears. This is the Third Noble Truth - cessation of dukkha.
However, because all circumstances depend on some conditions, and those conditions depend on other conditions - all circumstances are inevitably subject to cessation, whether we want it or not. So our idea of permanent peace is a big fiction. Permanent peace would need to not depend on any conditions in order to be permanent. Which is impossible. Hence, permanent peace is false. Things are impermanent. This is first Mark of Existence, "anicca".
Also, because things depend on conditions and always in flux, having "me" be always happy or always free would require a permanent "me" - otherwise the freedom will end with the end of the "me". Because "me" depends on causes-n-conditions, it is conditional just like everything else. There is no solid core that is "me", it is only a temporary arrangement. This is second Mark of Existence, "anatta".
All things are like this, only temporary arrangements. But for simplicity our mind builds a semi-static model of reality, in which it fabricates the notion of "objects", as if things existed statically. Because this is the way mind works, the stable objects it sees are only virtual constructs, conceptual overlays, on top of the soup that is always in flux. Because the things in flux always change, the mind always falls behind updating its models. So the mind model of reality is always mismatching the actual reality, at least to some degree. As we saw above, it is exactly this mismatch from which comes the dukkha or suffering. Therefore the mind that builds models of reality is always subject to suffering. This is third Mark of Existance, "dukkha".
When we understand the above, we understand to which degree we can have control over our experience, and how to minimize dukkha by dropping attachment to legacy or unrealistic models, and keeping our models in sync with reality as much as possible.
On the other hand, the mind generates joy when its model does match reality. In modern psychology this is known as the reward mechanism. When we know this we can hack the mechanism. We can generate joy by following these steps: 1) getting rid of course action that gets us in trouble (violence etc.), 2) then by getting rid of the three "klesha"s of attraction, aversion, and confusion (the coarse mismatch) - and their derivatives; and then 3) by contemplating the Noble Truths and three Marks of Existence and appreciating how lucky we are to have achieved such insight into the nature of phenomena and control over our experiences. This self-generated joy can give us some nice refuge to dwell in. This is First Jhana.
The Jhanas then naturally progress from more coarse and artificially generated happiness, to more subtle and natural happiness, then to calmness and surety. These are Second, Third, and Fourth Jhanas.
And then you realize that to be stuck in calmness is rather constricting - and allow yourself to get disenchanted and lose any form altogether. You are no longer defined by any strategy. You are done. But by having gone through all the previous stages you ensure you are not caught up in the coarser issues either.
At this stage the question of reality is no longer relevant. Reality is an interpretation we make. We choose the interpretation as needed to face the situation at hand.
So, as you see, desire of no desire is a very naive simplification. In practice it is not desire that we are working with, it is attachment, in its various forms, and the process goes through many stages from coarser to subtler until it gets thin enough to short circuit. This process is the Fourth Noble Truth, the path.
You can't skip the obstacles by dropping the path, the peace is not achieved by accepting the war.
Something like this.