  In this paper, we study the Euler transform on linear ordinary differential
operators on $\mathbb{P}^{1}$. The spectral type is the tuple of integers which
count the multiplicities of local formal solutions with the same leading terms.
We compute the changes of spectral types under the action of the Euler
transform and show that the changes of spectral types generate a transformation
group of a $\mathbb{Z}$-lattice which is isomorphic to a quotient lattice of a
Kac-Moody root lattice with the Weyl group as the transformation group.
