= Solution
Interpret stationarity in the usual second-order time-series sense and assume nondegenerate noise, $\sigma_z^2>0$. For a two-sided autoregressive equation, the missing existence condition is
$$
\boxed{|\phi|\ne1.}
$$
It is important to separate this from causality. If $|\phi|<1$, the unique stationary solution is $X_t=\sum_{j\geq0}\phi^jZ_{t-j}$. If $|\phi|>1$, there is still a stationary solution, but it is <anticausal>:
$$
\boxed{X_t=-\sum_{j=1}^\infty\phi^{-j}Z_{t+j}.}
$$
Both expansions converge in L2 because their coefficients are square summable. Substitution verifies the equation. Their means are zero and their <covariance> functions depend only on lag. Uniqueness follows by iterating the equation backward in the first case and forward in the second: the remainders $\phi^nX_{t-n}$ or $\phi^{-n}X_{t+n}$ tend to zero in L2 for any stationary finite-<variance> solution. This is the <stationary versus causal solution of a two-sided AR(1) equation>.
For $\phi=\pm1$, iteration gives
$$
X_t-\phi^nX_{t-n}=\sum_{j=0}^{n-1}\phi^jZ_{t-j}.
$$
The <variance> of the right side is $n\sigma_z^2$. The <variance> of the left side is at most $4\operatorname{Var}(X_t)$ by stationarity and <Cauchy-Schwarz inequality>. These are incompatible as $n\to\infty$. Thus no weakly stationary finite-<variance> solution exists at those <unit roots>.
If the intended claim includes a causal innovation representation, its condition is instead \b[$|\phi|<1$], as in the next part. The stated <white noise> equation alone does not say that $Z_t$ is orthogonal to the past of $X$. If zero innovation <variance> is allowed, the unit-root exclusion has degenerate exceptions, such as random constant solutions when $\phi=1$; the nondegenerate convention is necessary for the asserted nonexistence.
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