White Under Eye Rings from CO2 Erbium Laser?

Deep ablative laser resurfacing with CO2 or Erbium flat beam lasers damages pigment cells and causes hypopigmentation. The effect is worse when only one area of the face is treated, because the untreated skin maintains its normal color, and the treated area looks even whiter. The transition between treated and untreated skin is called a dermacation line and occurred in most agressive ablative flat beam laser resurfacing.
Newer fractional Erbium laser resurfacing is less likely to cause hypopigmentation.
One possible solution to your problem is to find a truely expert physician who has a lot of experience doing ablative laser resurfacing.
Staged, low power flat beam CO2 laser resurfacing can be done to reduce pigmentation in the lower face and lessen the demarcation line or difference around the eyes, by lightening the surrounding skin.
New non-ablative fractional Erbium 1540 laser treatments have improved pigmentation in some hypopigmented scars. This may be an option but you need to find an up to date expert in laser resurfacing

Subunit laser peels can often cause skin color changes that are noticeable. That is why, I prefer to laser the entire face. You can either use make-up to camouflage the area or attempt a full face peel of the other regions to match the color.

The problem you descibe is consistent with hypopigmentation, which is a sequelae of both CO2 and the erbium Lasers. The loss of pigment is usually permanent, however, you can camouflage it some what with either an Obagi peel or with a less aggressive fractional CO2 laser treatment.
I would consult with a physician who has experience with the later. It may take more than one treatment, but it will certainly help blend in the demarcation line.