President Reagan is undergoing a battery of annual health checkups, but White House officials say they'll keep the results private. Reagan, who is spending a week of his August vacation in Los Angeles with his wife, Nancy, had a checkup Thursday with his allergist, Dr. Ralph Bookman, White House spokesman Marlin Fitzwater said. The spokesman described the session as "his regular, annual allergy checkup," conducted in the privacy of the president's penthouse suite at the plush Century Plaza Hotel. The president was to undergo eye, ear, nose and throat examinations before returning to his ranch near Santa Barbara on Saturday, the spokesman said. Fitzwater said the president takes weekly shots for desensitization to pollen.