A laundry washing machine comprising an outer casing and, inside said outer casing, a washing tub, a rotatable drum housed in axially rotatable manner inside the washing tub and structured for housing the laundry to be washed, a detergent dispenser which is structured for supplying detergent into the washing tub, a fresh-water supply circuit which is structured for selectively channelling a flow of fresh water from the water mains towards the detergent dispenser and/or the washing tub, and an internal water softening device filled with a water softening agent capable of reducing the hardness degree of the fresh water directed towards the detergent dispenser or the washing tub; a regeneration-agent reservoir located/recessed inside the outer casing and structured for being manually fillable with a given amount of consumable salt or other regeneration agent; a first water-supply line which is structured for selectively channelling a flow of fresh water into said regeneration-agent reservoir so as to form some brine; and a brine reservoir which is discrete from said regeneration-agent reservoir, is dimensioned to catch and contain a given amount of brine and is fluidically connected to said regeneration-agent reservoir for receiving and accumulating the brine coming out from said regeneration-agent reservoir; the laundry washing machine being characterized by additionally comprising a detector assembly which is associated to the brine reservoir and is capable of detecting when the salinity degree of the brine stored into the brine reservoir exceeds a predetermined minimum salinity value and when the level of the water or brine stored inside the brine reservoir is equal to or higher than a predetermined first threshold value (L 0); said detector assembly including a salinity detector device that comprises: a first floating body which is housed inside the brine reservoir with the capability to move upwards and downwards and has a nominal density higher than that of the fresh water and lower than that of the brine having a salinity degree equal to said minimum salinity value, so as to float only in presence of selected brines having a salinity degree equal to or higher than said minimum salinity value; and a corresponding first electronic control unit capable of monitoring the position of said first floating body; said detector assembly moreover including a water-level detector device that comprises: a second floating body which has a nominal density lower than that of the fresh water, and is housed inside the brine reservoir with the capability to freely move upwards and downwards; and a corresponding second electronic control unit capable of monitoring the position of said second floating body. the laundry washing machine being characterized by additionally comprising a detector assembly which is associated to the brine reservoir and is capable of detecting when the salinity degree of the brine stored into the brine reservoir exceeds a predetermined minimum salinity value and when the level of the water or brine stored inside the brine reservoir is equal to or higher than a predetermined first threshold value (L 0); said detector assembly including a salinity detector device that comprises: a first floating body which is housed inside the brine reservoir with the capability to move upwards and downwards and has a nominal density higher than that of the fresh water and lower than that of the brine having a salinity degree equal to said minimum salinity value, so as to float only in presence of selected brines having a salinity degree equal to or higher than said minimum salinity value; and a corresponding first electronic control unit capable of monitoring the position of said first floating body; said detector assembly moreover including a water-level detector device that comprises: a second floating body which has a nominal density lower than that of the fresh water, and is housed inside the brine reservoir with the capability to freely move upwards and downwards; and a corresponding second electronic control unit capable of monitoring the position of said second floating body.