 Invertible ARMA factorization of an AR(1)-plus-noise process

Filtering the observations by $1-\varphi B$ gives covariance $A={\sigma }_z^2+(1+{\varphi }^2){\sigma }_w^2$ at zero, $C=-\varphi {\sigma }_w^2$ at lag one and zero elsewhere. Set $\nu =(A+\sqrt{A^2-4C^2})/2$ and $\vartheta =C/\nu$. Then $\nu (1+{\vartheta }^2)=A$, $\nu \vartheta =C$ and $|\vartheta |<1$. Applying $(1+\vartheta B{)}^{-1}$ defines actual white-noise innovations with variance $\nu$, proving an at-most-(1,1) ARMA representation without assuming Gaussianity.