Headache after Pilates workout

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I recently started doing Pilates with a certified instructor.  We use the reformer and the cadillac equipment.  I am doing great until I get up after having been flat on my back for most of the session which is an hour.  I sit up and have a pretty severe headache, mostly frontal.  I do not get headaches normally in everyday life.  I DO have heart and lung disease but am doing very well and exercise 5 days a week, using a recumbent bike, treadmill, arc trainer, resistance training etc.  I never have gotten a headache doing any of those.  This headache goes away in about 10 minutes.  I am on supplemental oxygen at night, because I drop my saturation at night and also when I lie flat sometimes even  awake.  I have never had this kind of headache before I went on oxygen though either.  
Anyone have any ideas as to why I have it?  Should I just quit the Pilates, which I do enjoy?  I also do have pretty low blood pressure sometimes, like under 90 systolic, but I usually feel just fine with it.  I hope someone with Pilates experience can help!  Thanks.