I suspect this is rather obvious once explained; I must be misunderstanding something.
From reading various answers here (such as this one), I understand that an LTI system must have zero output for zero input, which I think is equivalent to zero initial conditions or initial rest. To quote @MattL from the answer I referenced,
"This explains why a system with non-zero initial conditions can neither be linear nor time-invariant. A linear system must have a zero output for zero input"
In the textbook I am working through, one of the problems states "Find the response for the system described by the following difference equation with the initial conditions given", and the difference equation is as follows:
Here the input starts at $n=0$ yet $y[-1]=-3$. The textbook is asking the question in the context of LTI systems.
At first I thought there was a mistake in the question, but I have seen similar setups in answers here on stackexchange as well.
How is the above scenario compatible with zero initial conditions?