Addition of an ether-group-containing second solvent (tetrahydrofuran, THF) to the binary system water/linear poly(vinyl methyl ether) proved to have a profound influence on liquid/liquid phase relations and to cause a sharp decrease in solvent power of the mixed solvent as compared to water alone. A classic thermodynamic analysis of the phase diagrams showed the ternary system to exhibit two mechanisms of three-liquid-phase separation, both passing through a critical end point and one of them involving a heterogeneous double critical point. The ternary miscibility gap vanishes in a homogeneous double critical point at about 18 degreesC. Replacement of the linear PVME by its cross-linked variety changed the phase behavior in a fashion resembling that of the binary system water/cross-linked PVME. A third mechanism of three-liquid-phase separation, not involving a critical end point, is now operative, experimental indications for which are presented. Below 30 degreesC, the ternary system water/ THF/PVME is partially miscible and presents an example of co-nonsolvency, since both water and THF each mix in all proportions with the polymer in that temperature range. An explanation for the phenomenon given by Winnik et al. for the system water/methanol/poly(N-isopropylacrylamide) appears also to hold true for the present system.