  We study the integrability of the four-dimensional eighth-order nonlinear
wave equation of Kac and Wakimoto, associated with the exceptional affine Lie
algebra ${\mathfrak e}_6^{(1)}$. Using the Painlev\'{e} analysis for partial
differential equations, we show that this equation must be non-integrable in
the Lax sense but very likely it possesses a lower-order integrable reduction.
