I have done some MH ballast replacements before but never on an ADA Solar I.
I looked at the specs and so on for the light and it seems to be using a 150w HQI MH bulb. You will need to match the ballast to the bulb. You can not use something different.
I found a few options that might work for you. It looks like the ADA ballast is using connections built right into the ballast. But I can't tell exactly what they are. There is a good chance that they are proprietary and designed so they will only work with an ATA fixture.
CoralVue LuXcore Selectable Watt Ballasts | Reef Aquarium Metal Halide Lighting
This one might be directly compatible. But even it it's not you might be able to still use it.
Metal Halide 150W Electronic Ballast (Assembled) - Hamilton - Bulk Reef Supply
This one might also be a plug compatible replacement. Note the configuration of the output connection and it's pin placements.
IceCap 150W HID Electronic Metal Halide Ballast
This would likely work, but you can see it's a raw ballast. you will need to mount it in an electrical box, and then make all your connections. It's doable but a pain.
On thing you have going for you is that MH lighting and ballasts are fairly standard. Other parts can usually be made to work, although there many be some rewiring involved.
A final note. You are dealing with some serious power here. You don't want to fry yourself in the process. If I need to splice wires together, I do it using an electrical box, then I solder the connections (an optional step), use a wire nut to cover the bare wire, and use electrical tape to make sure the wire nut can't come off. Note if the electrical box is metal, be sure that's connected to ground too.