A well-known construction scheme for exact solutions of integrable equations is related to stationary equations for higher symmetries from a commutative subalgebra. If we use also the noncommutative symmetries then the resulting reduction lose the Liouville integrability property, but retains the Painlev\'e property. Such reductions include string equations related to classical symmetries such as the Galilean boost and the scaling. These solutions are not very well studied, but we know from the works by Suleimanov, Dubrovin and others that among them there are special solutions that describe asymptotics in the vicinity of the breaking point. In the talk, some recent results are presented that show that reductions obtained by use of master-symmetries may turn useful in studying another physically important regime, such as the decay of step-like solutions. Already published results are related to the KdV equation and the Volterra lattice; some new examples are also reported.