Tremblay–Turbiner–Winternitz system at integer index k : Polynomial algebras of integrals

An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index k was introduced in Tremblay, Turbiner, and Winternitz [J. Phys. A: Math. Theor. 42, 242001 (2009)] and was called in literature the Tremblay–Turbiner–Winternitz (TTW) system. Its eigenenergies are linear in quantum numbers and correspond to 2D harmonic oscillator with frequency ratio 1:k. In this paper it is conjectured that the Hamiltonian and both integrals of the TTW system have a hidden algebra g(k) - this was checked for k = 1, 2, 3, 4 - having a finite-dimensional representation spaces as the invariant subspaces. It is checked that for k = 1, 2, 3, 4 the Hamiltonian H, its two integrals I1,I2 and their commutator I12=[I1,I2] are four generating elements of the polynomial algebra of integrals of the order (k + 1): [I1,I12]=Pk+1(H,I1,I2,I12), [I2,I12]=Qk+1(H,I1,I2,I12), where Pk+1, Qk+1 are polynomials of degree (k + 1) written as superposition of the ordered monomials of H,I1,I2,I12. This implies that the polynomial algebra of integrals is a subalgebra of g(k). It is conjectured that this is true for any integer k.