My two cents: a proof for $n=3$ is given explicitly by Dubrovin, Fomenko and Novikov in [1], §15.2 pp. 138-142. The author explain also how to extend their proof to the case $n>3$ and leave the details as an exercise.

**Reference**

[1] Boris A. Dubrovin, Analtoly T. Fomenko and Sergey P. Novikov, *Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, and fields*,  translated by Robert G. Burns. 2nd ed. (English) Graduate Texts in Mathematics, 93, Berlin-Heidelberg-New York: Springer-Verlag, pp. xv+468 (1992), [MR1138462](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1138462), [Zbl 0751.53001](https://zbmath.org/?q=an%3A0751.53001)