Well, here's the problem.
The way the lighting system is wired is not the way the ballast indicates it should be wired on it's wiring diagram.
The ballast diagram shows all of the wires being used, but only two are used for each bulb in your system.
The sockets have four wires as well, but they are wired funny. Two of the wires connect back to the ballast, but rather than the other two doing the same thing, they are wired together through some sort of component wrapped in heat shrink tubing. It appears to be a capacitor.
What has happened here is that Odyssea has chosen to save money by engineering the wiring differently. By doing what they've done here, they save a 4 wire run from the ballast to the hood. If you look, the cable connecting the ballast to the hood is 4 wires... they'd have to use two of those or an 8 wire cable to connect the ballast as shown, so they "rigged" it.
Unfortunately, this means that there are no guarantees whatsoever that a replacement ballast will work in the same way, since the way they engineered it is more than likely dependant on that specific ballast and the way it drives the bulbs.
Whats more, I cannot find a ballast that measures the same as yours... and trust me, I looked everywhere for one. You're lucky I didn't feel like working today. I also looked at the universal ballasts, such as the Icecap, and they require a minimum of 6 wires from the ballast to the bulbs. That's better than 8... but still two too many.
I do, however, have a potential solution. What I'm not sure of is if it's within your comfort zone as far as wiring is concerned.
I found a Coralife 2 x 96w PC ballast unit. It's an entire external ballast unit, so it would replace everything you've taken pictures of... the ballast, chassis, everything.
However, it looks to have two separate cables, one for each lamp. This would mean that you'd have to drill your hood (canopy) and run the new cables into it, and abandon the present cable. You'd then have to cut the sockets loose in your existing unit and splice the wires onto the cables from this unit (this coralife system has square pin sockets, your system has straight 'inline' style so you'll have to retro them on).
Theoretically speaking, this should work, but it will require some modification and wiring to make it happen. I'm sorry man, I was hoping that I could find you a drop in replacement, but unfrotunately I cannot.
I also cannot guarantee that I'll be able to completely help you through it. I've never had an Odyssea fixture in front of me and I've never seen this coralife ballast kit in person either. Therefore, we' d be in the dark.
We're not allowed to link, but it's on Marine Depot's web site, item number: ES00938
Take a look and let me know what you think.