  We introduce an integrable two-component extension of the general heavenly
equation and prove that the solutions of this extension are in one-to-one
correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we
demonstrate that if the metrics in question are hyper-para-K\"ahler, then our
system reduces to the general heavenly equation. We also present an infinite
hierarchy of nonlocal symmetries and a recursion operator for the system under
study.
