Do I eliminate AMT by selling the shares during the calendar year of exercise or simply by selling the shares within one year after the exercise?
You eliminate the AMT by selling, gifting, or otherwise transferring ISO shares during the calendar year of ISO exercise.
Example: You exercise ISOs in January 2023 and sell the shares in December 2023.
If you sell the shares in the calendar year after the year of exercise, the spread at exercise is still part of the AMT income calculation, even if the sale occurs less than a year after the date of exercise (e.g. you exercise in November 2023 and sell the shares in May 2024).
See a related FAQ on the tax consequences when you sell ISO shares in the year you exercised them. Another FAQ and an article (7 ISO Strategies That Can Help Minimize Alternative Minimum Tax) explain why exercising ISOs at the start of the year can be a sound strategy.
Alert: The wash-sale rule can apply if you sell ISO stock at price lower than the market price was on your exercise date to avoid the AMT and then quickly repurchase the stock.