Summary: Given more than one locally stationary (LS) time series, this article describes a method to discover time-varying linear combinations of the LS series that are stationary. Systems for which this can occur are called costationary, and the associated time-varying linear combinations are called costationary vectors. Costationary systems are interesting for a number of reasons. The costationary vectors shed light on the nature and strength of a potentially interesting relationship between the LS series. The derived stationary series, which is the time-varying combination of the LS series, is often of independent interest and use. The article discusses why a spectral approach is often preferred to the time-domain and why costationary vectors need to be complexity constrained, and it also demonstrates an interesting error-correction formulae which shows how costationary systems must evolve to maintain stationarity in response to system shocks. We illustrate our methodology with two examples: one from asset allocation in financial portfolio construction and the other which mitigates intermittency in wind power management. In the former, a stationary synthetic asset is constructed using market index data and is shown to have superior Sharpe ratios to two established portfolio selectors. In the latter, power outputs from separate wind series are dynamically combined to provide a power output which has smaller intermittency than the individual inputs.