anime about a girl with a scar???

[danae]

I have this vague memory of an anime (Japanese I guess) about this girl with a scar on her abdomen which she got when her parents were shot to death. There is also something about dream manipulation (she sleeps and this cable from the ceiling manipulates her dreams).
I have this picture of her remembering what happened to her and her parents when she sees the scar in the shower or something.
I have been trying to find this for many years. I guess it was the 80s or perhaps the early 90s that I watched it ??
I would really appreciate if anyone could help.
thank you!