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Linear first order systems of partial differential equations of the form$\nabla f = M\nabla g,$ where $M$ is a constant matrix, are studied on vectorspaces over the fields of real and complex numbers, respectively. TheCauchy--Riemann equations belong to this class. We introduce a bilinear$*$-multiplication on the solution space, which plays the role of a nonlinearsuperposition principle, that allows for algebraic construction of newsolutions from known solutions. The gradient equations $\nabla f = M\nabla g$constitute only a simple special case of a much larger class of systems ofpartial differential equations which admit a bilinear multiplication on thesolution space, but we prove that any gradient equation has the exceptionalproperty that the general analytic solution can be expressed through powerseries of certain simple solutions, with respect to the $*$-multiplication.

Multiplication for solutions of the equation $\grad{f} = M\grad{g}$