Volkemon, I share your year of origin, 2006, buying mine used from Earl last year. It also has the single 12 volt battery, and the first summer we used it the house battery seemd to always be low in spite of driving, and finally died that fall. In taking it to a battery shop for a replacement, the owner decided to investigate the possible reason for poor performance. He found with the engine running the battery wasn't being charged, AHA ! Upon further investigation he found a cable near the battery with it's end taped up. We removed the tape, attatched it to the battery, started the engine, and voila !
House battery now charging.
Previous owner must have "taped it off" for some reason, maybe for winterizing ?
So check around good under there, maybe the cable got detatched like mine. If none present maybe it's just a matter of running a cable between the two ?