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We establish all normal extensions domains structure on the weighed Hilbert space Lα2(H,(0,1))for a formally normal minimal operators class defined by a degenerate conformable α-order differential-operator (Bu)(α)(t)+Au(t),α,t∈(0,1),where a positive selfadjoint B:H→Hhas a closed range, KerBis a nontrivial subspace and A:H→His a selfadjoint operator. In addition, it is given the analysis of the spectrum set for any normal extensions.