If something's unvarying, it stays exactly the same, with no change or variety. Unvarying rainy days can make a person dream of a warm, sunny beach.
Things that don't fluctuate or change are unvarying — this can be dull and monotonous, as with a boring professor's unvarying lectures, or simply reliable, like your best friend's unvarying kindness. If you wear the same clothes every day, you can talk about your unvarying outfit, and if a radio station plays an identical mix of songs day in and day out, that's unvarying too. To vary is to change, from the Latin root varius, "varied, different, or spotted."