A note on elliptic first order systems in the plane and the Vekua Equation with structure polynomial $X^2 + \beta X + \alpha $

TL;DR

This paper investigates conditions under which elliptic first order systems in the plane can be reformulated as Vekua-type equations with parameter dependence, enhancing understanding of their structure and solution methods.

Contribution

It provides new criteria for transforming elliptic systems into Vekua equations with structure polynomial $X^2 + eta X + oldsymbol{ extalpha}$, expanding the theoretical framework.

Findings

Derived conditions for complex rewriting of elliptic systems

Established links between elliptic systems and Vekua equations with structure polynomial

Enhanced methods for analyzing elliptic first order systems in the plane

Abstract

This note deals with the following problem: under which conditions can elliptic first order systems in the plane be complex-rewritten as a parameter-depending Vekua-type equation?