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Using older and recent results on the integrability of two-dimensional (2d) dynamical systems, we prove that the results obtained in a recent publication concerning the 2d generalized Ermakov system can be obtained as special cases of a more general approach. This approach is geometric and can be used to study efficiently similar dynamical systems.
The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the mbox{$mathfrak{o}(2)^{oplus n}$} subalgebra of $mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $mathfrak{o}(2n)$. This result is shown to be rela
We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q)dot{q}^{a}dot{q}^{b}+K_{a}(t,q)dot{q}^{a}+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations which involve the
The newest model for space-time is based on sub-Riemannian geometry. In this paper, we use a combination of Lorentzian and sub-Riemannian geometry, the suggest a new model which likes to its ancestors, but with the most efficient in application. In c
In this paper we present a criterion for the covering condition of the generalized random matrix ensemble, which enable us to verify the covering condition for the seven classes of generalized random matrix ensemble in an unified and simpler way.
This paper is a natural continuation of the previous paper cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=alpha/x^2$, $alphageq-1/4$, were constructed. In this paper, we present generalized o