On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643-2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69-115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke's conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643-2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.