Consider a small, one- or two-seater motor GA aircraft, something like a Cessna 152 or similar, designed for level flight at roughly neutral pitch.
It's certainly possible to fly such an aircraft in a nose-high attitude while maintaining altitude, by careful use of pitch and thrust. I'm not sure I'd want to do that for any significant length of time, but as long as the angle of attack is kept below the critical angle for the current aircraft configuration, I can see no aerodynamic reason why it can't be done even in the general case.
But is it possible to fly it in a significant nose-down (say, several degrees or more) attitude while maintaining altitude?
My gut feeling says no; no matter what else you do with the aircraft in such a nose-down attitude, you're going to be losing altitude. But I've been surprised before, so: is there a way?