  The non-commutative harmonic oscillator (NCHO) was introduced as a specific
Hamiltonian operator on $L^2(\mathbb{R})\otimes\mathbb{C}^2$ by Parmeggiani and
Wakayama. Then it was proved by Ochiai and Wakayama that the eigenvalue problem
for NCHO is reduced to a Heun differential equation. In this article, we
consider some generalization of NCHO for $L^2(\mathbb{R}^n)\otimes\mathbb{C}^p$
as a rotation-invariant differential equation. Then by applying a
representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq
\mathfrak{su}(1,1)$, we check that its restriction to the space of products of
radial functions and homogeneous harmonic polynomials is reduced to a
holomorphic differential equation on the unit disk, which is generically
Fuchsian.
