I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we conclude that $l\in\mathbb{Z}^+$. My question is to know why this happens in a geometric and / or qualitative way. In other words, I want to know whether, in a qualitative or geometric way, it is possible to conclude that $l\in \mathbb{Z}^+$.



Legendre equation: An interpretation
I am not sure if this is the qualitative/geometric interpretation -of the integrality of the $l$ parameter- you are looking for, but if the parameter $l$ is a non-negative integer then the Legendre polynomials $P\_l$ and the associated Legendre polynomials $P\_l^m$ are proportional to the matrix elements of special cases of [Wigner's $D$-matrix](https://en.wikipedia.org/wiki/Wigner_D-matrix), which provides irreducible representations of the Lie groups $SU(2)$ and $SO(3)$. 


In other words, the physical significance (or the geometric interpretation if you prefer) of the integrality of the parameter $l$ is directly related to the fact that the corresponding solutions $P\_l$ (of legendre's DE) are (bounded) eigenfunctions of the Laplacian on $S^2$ and they provide irreducible, spin-$l$, representations of the Lie group $SO(3)$. 


