We address two closely related problems associated with the singlet scalars potential that are often present in supersymmetric U(1) models, especially those which maintain the gauge unification of the MSSM in a simple way. The first is the possibility of an accidental global symmetry which results in a light Goldstone boson. The second is the problem of generating a vacuum expectation value for more than one field without reintroducing the $mu$ problem. We give sufficient conditions for addressing both issues and provide a concrete example to generate them.
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