I brought my bike back home and unloaded it. I pulled the rear wheel off and yanked the tire off of the rim for an inspection. I was sure that I would find something causing the flats. After careful inspection of the inside of the tire, the rim and all of the spoke nipples, I came up with nadda, zip, zilch. The only thing I can think of is that, where the spike went into the tire (first flat), it left a pretty fair sized hole in the tire, and occasionally, a small rock will catch the hole in the tire just right and protrudes through and punctures the tube.
I cleaned the rim and tire up, put a new rim band on the rim, double patched the hole in the tire and mounted it back on the rim. I'm not sure how the patches will hold on the inside of the tire, I've never tried this before, but the tire only had 100 miles on it. Being a new Dunlop 606, I couldn't just peel it off and trash it without trying this. I am taking it out Tuesday for a test run ... hopefully with no flats on this ride.