LTI system and initial conditions 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:

$$y[n]-0.7y[n-1]=u[n], y[-1]=-3$$

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?
The system is described by a linear difference equation with constant coefficients and as such, it is described in the same way as a linear time-invariant system. It is just the non-zero initial condition that makes the system non-linear, at least according to the common definition of linearity in system theory (homogeneity and additivity).

Such a system with non-zero initial conditions is also referred to as incrementally linear. An incrementally linear system responds linearly to changes in the input. For this reason it is very common, and it also makes a lot of sense, to discuss it in the same chapter as general discrete-time LTI systems.

As a final note, in practice we're mainly interested in stable systems. For such systems, the influence of non-zero initial conditions becomes negligible after some time, and, consequently, the system will practically behave as an LTI system after the transient caused by the non-zero initial conditions has died out.