We study the coherent motion of bubbles and spikes in the Richtmyer-Meshkov instability for isotropic three-dimensional and two-dimensional periodic flows. For equations governing the local dynamics of the bubble, we find a family of regular asymptotic solutions parametrized by the principal curvature at the bubble top. The physically significant solution in this family corresponds to a bubble with a flattened surface, not to a bubble with a finite curvature. The evolution of the bubble front is described and the diagnostic parameters are suggested.