Laundry machine has a detergent dispensing assembly, a fresh-water supply circuit for selectively channeling a fresh water flow towards the detergent dispensing assembly, and a water softening device including a softening container having softening agent along the fresh-water supply circuit. A regeneration-agent reservoir fluidically connected to the softening container receives a regeneration agent. A water supply circuit channels fresh water into the regeneration-agent reservoir. The softening device includes an automatic one-way valve on a first fresh-water supply line upstream of the softening container, allowing water to only flow along the first fresh-water supply line, from a water mains to the softener container. An electrically-controlled valve assembly/flow diverter, located along the first fresh-water supply line, downstream of the one-way valve and upstream of the softening container, selectively allows/prevents the water to freely flow towards, respectively, the softening container and the soften
