Let $Y_{t}$, for $t=1,2, \dots$ be AR(1) process. $$ Y_{t+1} = c_{1} + \phi Y_{t} + \varepsilon_{t} $$
Next, assume that for some $X_{t}$ we have $$ X_{t} - \beta Y_{t} = u_{t}, $$ where $u_{t}$ is stationary. Therefore, we can say that $X_{t}$ and $Y_{t}$ are cointegrated.
Is it possible that $X_{t}$ is also AR(1)?
Attempt:
Assume that $u_{t}$ is i.i.d. Then, in this simplest case $X_{t}$ is $$ X_{t} = \beta Y_{t} + u_{t}, $$ therefore, it is ARMA(1,1). Though, it is not a proof of the original statement.