Are those with separate ballast and igniter? I suspect it has ballast+igniter combo. You could also figure the ballast by the type of bulb: it could be either D1S or D2S. D1S has ignitor in the base, D2S is fed by the combo. I would think that standard Philips ballast with the built-in ignitor would suffice, if it is the D2S. If the assembly has a dust cover on the back, you could reach and swap bulbs/ballasts with the healthy light to diagnose.
You can find the genuine Philips ballasts very cheap (compared to dealers' prices), or you could risk even cheaper fakes. If you need just a bulb, replace both as the color will be demonstrably different, as they change color temps with age.
All in all, it is not as bad as it used to be in terms of pricing nowadays.
Bulbs can be had on Amazon, Osram is about $40-$50, Philips a little more.