We report an analysis to the problem of nonlinear motion of bubbles and spikes generated by the Richtmyer-Meshkov instability. The flow is three-dimensional (3D), periodic and anisotropic in the plane normal to the direction of shock. We show that in the traditional Layzer-type approach, regular asymptotic solutions to the problem are absent in the general case. We propose yet another approach and find a family of regular asymptotic solutions parameterized by the principal curvatures at the bubble top. In the expanded functional space the interplay of harmonics is well captured. For solutions of this family, a bubble with a flattened surface is faster than a bubble with finite curvatures in both 3D and two-dimensional (2D) cases, while highly symmetric 3D bubbles are faster than anisotropic 3D and 2D bubbles. For nearly symmetric 3D flows, the Layzer-type solution is the point of bifurcation. (C) 2001 American Institute of Physics.