Monogenic Functions with Values in Commutative Algebras of the Second Rank with Unit and the Generalized Biharmonic Equation with Double Characteristic

We prove that any commutative and associative algebra 𝔹* of the second rank with unit over the field of complex numbers ℂ contains bases {e1, e2} for which 𝔹*-valued “analytic” functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differentional equation of the fourth order with complex coefficients whose characteristic equation has a single multiple root and the remaining roots are simple. We present a complete description of the set of all triples (𝔹*, {e1, e2}, Φ).