In this article the authors give the following expansion of wavefunction of three-body system (equation $(16)$ in text):
$$\Psi(\textbf{x},\textbf{y})=\sum_{q=\lambda}^l\psi_q(\textbf x^2,\textbf y^2,\textbf x\cdot\textbf y)Q_q(\textbf{x},\textbf{y}),$$
where $\textbf x$ and $\textbf y$ are scaled Jacobi coordinates (the third one is removed as center of mass position), $Q_q(\textbf{x},\textbf{y})$ are the eigenfunctions of $\hat L^2$, $\hat L_z$ and some other momentum-related operators (these functions are related to Wigner D-matrix, but expressed in non-angular coordinates), and $\psi_q(\textbf x^2,\textbf y^2,\textbf x\cdot\textbf y)$ are functions of the scalar coordinates, i.e. independent of system orientation.
What I don't understand is: why is it necessary to use the sum of products of angular-dependent functions with angular-independent ones, and not just one product as the solution? As I know from two-body problem, there one separates spherical harmonics from the radial function, and doesn't do any sum over harmonics with the same $l$, for example. Also, for the case of three two-dimensional particles there's no such feature, because the total angular momentum has only one dimension in this case. What's different here?
Are there any examples of PDEs where the solution needed a similar non-trivial separation of variables, but there were fewer dimensions, so that it could be visualized more easily?