  We introduce a new family of $N$-dimensional quantum superintegrable model
consisting of double singular oscillators of type $(n,N-n)$. The special cases
$(2,2)$ and $(4,4)$ were previously identified as the duals of 3- and
5-dimensional deformed Kepler-Coulomb systems with $u(1)$ and $su(2)$ monopoles
respectively. The models are multiseparable and their wave functions are
obtained in $(n,N-n)$ double-hyperspherical coordinates. We obtain the
integrals of motion and construct the finitely generated polynomial algebra
that is the direct sum of a quadratic algebra $Q(3)$ involving three
generators, $so(n)$, $so(N-n)$ (i.e. $Q(3)\oplus so(n) \oplus so(N-n)$ ). The
structure constants of the quadratic algebra themselves involve the Casimir
operators of the two Lie algebras $so(n)$ and $so(N-n)$. Moreover, we obtain
the finite dimensional unitary representations (unirreps) of the quadratic
algebra and present an algebraic derivation of the degenerate energy spectrum
of the superintegrable model.
