New biharmonic bases in commutative algebras of the second rank and monogenic functions related to the biharmonic equation

Among all two-dimensional commutative algebras of the second rank a totally of all their biharmonic bases $\{e_1,e_2\}$, satisfying conditions ${(e_1^2+e_2^2)}^2=0$, $e_1^2+e_2^2\mathrm{\ne }0$, is found in an explicit form. A set of “analytic” (monogenic) functions satisfying the biharmonic equation and defined in the real planes generated by the biharmonic bases is built. A characterization of biharmonic functions in bounded simply connected domains by real components of some monogenic functions is found.