Abelian Equations and Rank Problems for Planar Webs
  We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincar\'{e}'s theorem: a planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem ``to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable. 
Vladislav V. Goldberg (New Jersey Institute of Technology, USA) and   Valentin V. Lychagin (University of Tromso, Norway)