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In this talk,we study the interlacing properties of zeros of para-orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dμ and para-orthogonal polynomials associated with a modification of dμ by the addition of a pure mass point,also called Uvarov transformation.Moreover,as a direct consequence of our approach,we present some results related with the Christoffel transformation.Some numerical experiments are presented.