Cleaning system for the reduction of SOx and particulate matter in exhaust gases from a marine combustion engine, burner or boiler, the cleaning system comprising: a first scrubber process loop comprising a first scrubber and a first water circulation tank, wherein water from the first water circulation tank is arranged to circulate in the first scrubber process loop, the first scrubber being arranged to receive exhaust gases from the combustion engine, burner or boiler and water from the first water circulation tank, which water, at least partly, is arranged to be evaporated to water vapor inside the first scrubber by contact with the exhaust gases, whereby the water vapor and the exhaust gases form humid exhaust gases; a second scrubber process loop comprising a second scrubber and a second water circulation tank, wherein water from the second water circulation tank is arranged to circulate in the second scrubber process loop, a communication between the first and the second scrubber (102, 202) allowing transfer of the humid exhaust gases from the first scrubber to the second scrubber; the second scrubber being arranged to receive water from the second circulation tank, which water is arranged to condensate, inside the second scrubber, the water vapor in the humid exhaust gases at least partly; and a communication (300; 300') between the first and the second scrubber process loop (101, 201) allowing a reflux of water from the second scrubber process loop to the first scrubber process loop, the cleaning system being characterized in that it further comprises an arrangement for supply of an alkaline agent to at least the second scrubber process loop, wherein at least 60 % and more preferred at least 90 % of the total amount of the alkaline agent supplied to the cleaning system is supplied to the second scrubber process loop. a first scrubber process loop comprising a first scrubber and a first water circulation tank, wherein water from the first water circulation tank is arranged to circulate in the first scrubber process loop, the first scrubber being arranged to receive exhaust gases from the combustion engine, burner or boiler and water from the first water circulation tank, which water, at least partly, is arranged to be evaporated to water vapor inside the first scrubber by contact with the exhaust gases, whereby the water vapor and the exhaust gases form humid exhaust gases; a second scrubber process loop comprising a second scrubber and a second water circulation tank, wherein water from the second water circulation tank is arranged to circulate in the second scrubber process loop, a communication between the first and the second scrubber (102, 202) allowing transfer of the humid exhaust gases from the first scrubber to the second scrubber; the second scrubber being arranged to receive water from the second circulation tank, which water is arranged to condensate, inside the second scrubber, the water vapor in the humid exhaust gases at least partly; and a communication (300; 300') between the first and the second scrubber process loop (101, 201) allowing a reflux of water from the second scrubber process loop to the first scrubber process loop, the cleaning system being characterized in that it further comprises an arrangement for supply of an alkaline agent to at least the second scrubber process loop, wherein at least 60 % and more preferred at least 90 % of the total amount of the alkaline agent supplied to the cleaning system is supplied to the second scrubber process loop.