Commuting-flow symmetries and common solutions to differential equations with common symmetries

We point out that in certain cases, all the differential equations (for given indipendent and dependent variables) possessing a given symmetry necessarily share a common solution. Under weaker conditions, all such differential equations have a solution—in general, different for different equations—characterized by a common symmetry. We characterize this situation and the common solution, or the common symmetry of solutions, and give concrete examples.