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The answer is it depends.
Do you live in a community with an association? We would need to take a look at the bylaws of the association to determine for certain who is in charge of what. However, typically, the association is in charge of all common areas (and typically the sprinkler system is something that is installed throughout the entire property and not on an individual basis).
That being said, if the sprinkler system is your sprinkler system and does not belong to the general property, then you would be on the hook. If the sprinkler system belongs to the association, it is the responsibility of the property manager to fix it and you can sue the company in small claims court for the increase in the water bill.