Mistresses and illegetimate children of Emperor Franz Josef

My paternal grandmother and her family emigrated to Connecticut from Hanidowa, Ukraine in 1913. At the time, the region was part of Austria-Hungary. They were Oborskis; my grandmother's name was Sozha Oborska. She was born in 1897. Her father's name was Steven.

During a visit to my grandmother's house in 1975, I told her I was reading "Nicholas and Alexandra." I related what I'd been reading about; she listened intently, interrupting only now and then to correct my pronunciation of family & place names. Finally, I asked her if she had ever seen any of the Romanovs.

   "No, honey," she replied, "We live in small Ukrainian town. Sometimes it's Ukrainia, sometimes it was Poland. When we live dere, it was Austria."

   "Then did you ever see emperor Franz Josef?" I asked, harking back to a passage I'd read in Massey's book, about a visit paid by N&A to the old emperor.

   "Yes," she answered. Suddenly, her mood became more serious. "Franz Josef was my father, you great grandfather's uncle."

   I was stunned. "When was the last time you saw him?" was all I could think of to ask.

   "At my grandmother, you great great grandmother's funeral," she replied in her thick accent.

   "What was her name?"

   "Olga, honey," she answered, "Olga Oborska. When my father was born, she keep her last name and give it to him."

   "Who was your father's father?" I asked, completely fascinated.

   "Ludwig. It was Karl Ludwig. But dey not married," she added, "dat's why you great great grandmother keep her last name. When my father was born, Karl Ludwig give her dat farm so she could live."

   "Did you ever see Franz Ferdinand and Sophie Chotek?" I asked, thoroughly fascinated and groping for questions.

   "Yes, I see dem from time to time. Dey still family. Sophie was like older sister to me before she die," she added with a faraway look in her eyes, "she give me dresses."

   There was a pause. I cast about for more questions, but my grandmother leaned closer to me and patted me on the knee. "Don't you tell nobody, honey," she said firmly.

   "Why?" I asked.

   "Because I say so. We better off dat way."

   And that was it. I didn't ask her about it again. She died three years later, in 1978. The following year, I asked her daughter, my elderly Aunt Anna, "Who was Gramma's grandfather?"

   "Karl Ludwig," she said, "Why do you ask?" When I told her about the conversation I had had with my grandmother, she said, "Honey, that was all so long ago. Your grandmother never liked to discuss it, nor do I. You keep whatever you discussed with your grandmother to yourself."

   And so I have, for 36 years. I still wonder about it, and have many questions. I wouldn't hesitate to ask about my ancestry now, but my grandmother and her children - the two generations who would know the answers to my questions - are long gone. I know that the Hapsburgs sired many illegitimate children in the latter half of the 1800s and that they didn't abandon them, but I can't imagine that records exist about illegitimate children born to that House. Or do they? If so, where would I look?