Are discrete systems defined by LCCDE always LTI? Suppose a discrete-time system is defined by linear constant-coefficient difference equation

$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$

where at least two different coefficients $a_i,a_j$ are nonzero (so the equation is recursive). Isn't it true that such system is LTI and causal regardless of any auxiliary condition? After all, it is causal because it does not depend on any future inputs (RHS of the equation has only present or past inputs). And it is LTI because: if $x_1[n] \to y_1 [n]$ and $x_2[n] \to y_2 [n]$ the above difference equation is valid for $x_1,y_1$ and $x_2,y_2$ and by adding them together we can show that

$$\sum_{k=0}^{N} a_k (y_1[n-k]+y_2[n-k]) = \sum_{k=0}^{M} b_k (x_1[n-k]+x_2[n-k])$$

which means $x_1[n]+x_2[n] \to y_1[n]+y_2[n]$.

(similarly we can prove the homogeneity and the time-invariance property).

However it doesn't seem to be true according to Signals &amp; Systems by Oppenheim A.:


  As in the continuous-time case, such equation does not completely specify the output in terms of the input. To do this, we must also specify some auxiliary conditions.


Up to this point I agree. But then they write:


  [..] we will focus for the most part of the condition of initial rest - i.e., if $x[n] = 0$ for $n &lt; n_0$, then $y[n]=0$ for $n&lt;n_0$ as well. With initial rest, the system described by the [first] equation is LTI and causal


In other words, the author seems to imply that without the initial rest condition, there's no guarantee that the system is LTI and causal (why contradicts my previous statements). Where's my mistake?
What you have is a so-called behavior which truncates the signal space $\mathcal{X}\oplus\mathcal{Y}$ of all possible signal pairs $(x, y)$ to the ones that satisfies this equality. 

This does not specify an input-output mapping and hence a dependence and hence causality does not apply here. They simultaneously satisfy this. However regardless of causality, it is LTI since you can take arbitrarily and superposition applies.

Just like you can integrate any differential equation sufficiently many times and end up with a integral equation, you can also forward time shift these equations then if one of them depends on future inputs and that is your choice of dependent variable then you can speculate about causality.

This is actually the tool to allow for nondifferentiable input signals and other discontinuities and the results are investigated under the name of weak solutions