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abstract: 'In the previous works [@N46; @N47] authors have defined the oscillator-like system that associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev - Koornwinder oscillator. In this paper we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev - Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space $\mathcal{H}$ of functions that are defined on a region which bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev - Koornwinder polynomials and these polynomials form an orthonormalized basis in the space $\mathcal{H}$. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.'
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**V.V.Borzov$^1$, E.V.Damaskinsky$^2$**

$^1$Department of Mathematics, St.Petersburg University of Telecommunications, 191065, Moika 61, St.Petersburg, Russia; borzov.vadim@yandex.ru

$^2$Department of Natural Sciences, Institute of Defense Technical Engineering (VITI), 191123, Zacharievskaya 22, St.Petersburg, Russia; evd@pdmi.ras.ru

**The algebra of two dimensional generalized**

**Chebyshev - Koornwinder oscillator [^1]**

INTRODUCTION {#Intro}
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The notion of the quantum harmonic oscillator is one of the cornerstones of quantum physics. 