It is well-known that the zeros of holomorphic functions in more than one complex variable are not isolated. Nevertheless, there exist so-called identity surfaces such that a holomorphic function vanishes identically everywhere if only it equals zero on an identity surface. In the paper identity surfaces will be constructed using the technique of completely integrable overdetermined systems of partial differential equations. Moreover, identity surfaces will be constructed not only for holomorphic functions but also for solutions of more general first order systems of partial differential equations.The present paper deals only with systems with real-analytic coefficients and, therefore, the classical Cauchy-Kovalevskaya and Holmgren theorems are applicable (while many recent papers deal with infinitely differentiable coefficients or they solve initial value problems with generalized analytic initial functions). Using the compatibility conditions of an overdetermined system, in the present paper the construction of identity surfaces (of minimal dimension) is carried out as some kind of inverse problem to an initial value problem.