For each simple euclidean Jordan algebra $V$ of rank $ρ$ and degree $δ$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $ν$, is obtained. Here, $ν\in{\mathcal W}(V):=\{k {δ\over 2}\mid k=1, ..., (ρ-1)\}\cup((ρ-1){δ\over 2}, \infty)$ and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of $V$.
For the quantum dynamic problem labelled by $ν$, the bound state spectra is $-{1/2\over (I+ν{ρ\over 2})^2}$, I=0, 1, ... and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by $ν$. A few results in the literature about these representations become more explicit and more refined.
The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: ${1\over 2} ||\dot x||^2+{1\over r}$. Here, $\dot x$ is the velocity of a unit-mass particle moving on the space consisting of $V$'s semi-positive elements of a fixed rank, and $r$ is the inner product of $x$ with the identity element of $V$.
Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（博戈留博夫理论物理实验室，俄罗斯杜布纳联合核研究所）；Yerevan State University（埃里温国立大学）；Yerevan Physics Institute（埃里温物理研究所）；Institute of Radiophysics and Electronics（辐射物理与电子研究所）
Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
最近，通过找到一个附加的（隐藏的）动量四次积分，在三维欧几里得空间上叠加有三个离心项的开普勒-库仑势被证明是最大超可积的[Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902]。在本文中，我们提出的推广，这一结果的ND球面，双曲和欧几里德空间，利用统一的对称性的方法，利用曲率参数。由此产生的哈密顿量，形成的（弯曲）开普勒-库仑势连同N离心条款，被赋予（2N-1）功能独立的运动积分：其中一个是四次和其余的是二次。从适当的开普勒-库仑势，其相关的二次Laplace-Runge-Lenz N-向量，广义系统的过渡得到了充分的描述。在所有这些系统中的球形，非线性（立方），和余代数对称性的作用是突出的。
Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
在广义Kustaanheimo-Stiefel（KS）变换的框架下，给出了对偶（拟）精确可解模型及其隐对称代数的个数在不同分析层次上的描述.证明了N维奇异振子与（n+1）维广义MICZ-Kepler系统是对偶的，其对偶变换是KS变换的推广.在二重坐标系、球坐标系和抛物坐标系下，用分离变量法给出了这些问题的薛定谔方程的可解性。在SU（1，1）代数加法规则的框架下，二次Hahn代数QH（3）作为一个隐对称，不随原实空间${\rm I \!R}^{N}$分解成不同分量而改变.在Howe对偶的意义下，用交换子方法也清楚地证明了Higgs/Hahn代数等隐对称代数。对二维奇异振子进行降维处理，其中每个变量都是n维超半径r。本文还讨论了（N=2n）维奇异振子与（n+1）维广义MICZ-Kepler系统在拟精确（QE）问题类中的对偶联系。在广义KS变换的框架下，给出并分析了各向异性和非线性非谐项对谐振子模型及其对偶模型的某些推广。本文讨论了上述问题的薛定谔方程在QE类中的精确解析解，并给出了四类对偶拟精确可解模型的精确解析解。特别地，给出了在低维情况下与类似结果的比较及其推广.
MICZ-Kepler轨道是MICZ Kepler问题（Kepler问题的磁化版本）的非碰撞轨道。定向的MICZ-Kepler轨道可以由正则角动量$\mathbf L$和Lenz矢量$\mathbf A$参数化，参数空间由3D矢量$(\mathbf A, \mathbf L)$和${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$组成。最近的开普勒问题的4D视角产生了一个新的参数化，参数空间由闵可夫斯基向量对$(a,l)$与$l\cdot l =-1$，$a\cdot l =0$，$a_0>0$组成。这种新的轨道参数化意味着不同磁荷的MICZ-开普勒轨道通过对称性相互关联：\emph{${\mathrm {SO}}^+(1,3)\times {\mathbb R}_+$ acts transitively on both the set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic MICZ-Kepler orbits}。这个作用扩展到${\mathrm {O}}^+(1,3)\times {\mathbb R}_+$，即秩为2的欧氏若当代数的\emph{structure group}，该代数的洛伦兹空间是闵可…
