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Here are described the axiumbilic points that appear in generic one parameterfamilies of surfaces immersed in R4. At these points the ellipse of curvatureof the immersion, Little, Garcia - Sotomayor has equal axes.  A review is made on the basic preliminaries on axial curvature lines and theassociated axiumbilic points which are the singularities of the fields ofprincipal, mean axial lines}, axial crossings and the quartic differentialequation defining them.  The Lie-Cartan vector field suspension of the quartic differential equation,giving a line field tangent to the Lie-Cartan surface (in the projective bundleof the source immersed surface which quadruply covers a punctured neighborhoodof the axiumbilic point) whose integral curves project regularly on the linesof axial curvature.  In an appropriate Monge chart the configurations of the generic axiumbilicpoints, denoted by E3, E4 and E5, are obtained by studying the integral curvesof the Lie-Cartan vector field.  Elementary bifurcation theory is applied to the study of the transition andelimination between the axiumbilic generic points. The two generic patternsE^1_{34} and E^1_{45} are analysed and their axial configurations are explainedin terms of their qualitative changes (bifurcations) with one parameter in thespace of immersions, focusing on their close analogy with the saddle-nodebifurcation for vector fields in the plane .  This work can be regarded as a partial extension to R4 of the umbilicbifurcations in Garcia - Gutierrez - Sotomayor for surfaces in R3. With lessrestrictive differentiability hypotheses and distinct methodology it has pointsof contact with the results of Gutierrez - Guinez - Castaneda.

Axiumbilic Singular Points on Surfaces Immersed in R4 and their Generic Bifurcations