Painleve test integrability of nonlinear Klein-Fock-Gordon equations

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Abstract
The applicability of the Painleve test of the complete integrability of the one-component nonlinear Klein-Fock-Gordon equations in an arbitrary Riemannian space, in the formulation of Weiss et al. (J. Math. Phys., vol.25, p.13, 1984) is discussed. Three infinite series of these equations are found in the flat two-dimensional space which possess the Painleve property and include, as a special case, the Liouville, sine-Gordon, and Dodd-Bullough equations. It is pointed out that the approach of Weiss et al. to select integrable nonlinear equations is not sufficiently reliable and needs some strengthening.