Multiplication for solutions of the equation gradf=Mgradg

Linear first-order systems of partial differential equations (PDEs) of the form ∇f=M∇g∇f=M∇g, where MM is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy–Riemann equations belong to this class. We introduce on the solution space a bilinear ∗∗-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation ∇f=M∇g∇f=M∇g is a simple special case of a large class of systems of PDEs, admitting a ∗∗-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as ∗∗-power series of certain simple solutions.