The Cauchy–Riemann equations admit a bilinear multiplication of solutions, since the product of two holomorphic functions is again holomorphic. This multiplication plays the role of a nonlinear superposition principle for solutions, allowing for construction of new solutions from already known ones, and it leads to the exceptional property of the Cauchy–Riemann equations that all solutions can locally be built from power series of a single solution z = x + iy ∈ C.
In this thesis we have found a differential algebraic characterization of linear first order systems of partial differential equations admitting a bilinear ∗-multiplication of solutions, and we have determined large new classes of systems having this property. Among them are the already known quasi-Cauchy–Riemann equations, characterizing integrable Newton equations, and the gradient equations ∇f = M∇g with constant matrices M. A systematic description of linear systems of PDEs with variable coefficients have been given for systems with few independent and few dependent variables.
An important property of the ∗-multiplication is that infinite families of solutions can be constructed algebraically as power series of known solutions. For the equation ∇f = M∇g it has been proved that the general solution, found by Jodeit and Olver, can be locally represented as convergent power series of a single simple solution similarly as for solutions of the Cauchy–Riemann equations.