超可积系统与Zoll旋转度量
superintegrable systems versus Zoll metrics of revolution
摘要：
Koenigs构造了一族在动量中具有一个线性积分和两个二次积分的二维超可积（SI）模型，简称（1，2）。最近Matveev和Shevchishin表明，这种结构确实推广到具有一个线性和两个三次积分的模型，即$(1,3)$，直到非线性常微分方程的解。我们对这个方程的显式解使得可以构造这些SI系统，并证明了在S^2上全局定义的系统是Zoll系统。我们将把这些结果推广到（1，n）的情形，对于任意n个\geq 2.我们的方法也是构造性的，当n是奇数时，我们发现存在全局定义在S^2上的度量，这些度量确实是Zoll的（在参数的适当限制下），而如果n是偶数，我们发现的度量从来不是全局定义在S^2上的，就像柯尼希斯构造的（1，2）模型的情况一样。
英文摘要：
Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. $(1,3)$, up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S^2 are Zoll. We will generalize these results to the case (1,n) for any n \geq 2. Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S^2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S^2, as it is already the case for the (1,2) models constructed by Koenigs.
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