The paper presents two new proofs of the unsolvability of certain partial differential equations of first order \(Lf=g\). Both proofs depend on the theory of holomorphic functions of several complex variables. In the first case the unsolvability of \(Lf=g\) results from the existence of peak points in the topological algebra that is the kernel of \(L\). The proof yields a removable singularities theorem for \(L\). The second proof depends on the extension property of Hartogs type.
[See H. Lewy, Ann. Math. (2) 64, 514–522 (1956; Zbl 0074.06204).]