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The unitary extension principle 
$\left( \text{UEP} \right)$ by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the 
$\text{UEP}$-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one $\text{UEP}$-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.
