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A continuous counterpart of the spatial images technique is proposed for the computation of the multilayered boxed Green's functions and their derivatives. The method employs a set of auxiliary linear distribution of sources to effectively impose the potential boundary conditions along the whole cavity contour. The imposition of these boundary conditions leads to a set of integral equations (IEs), on the unknown distributions of the auxiliary sources, which are solved by applying a method of moments approach. A convergence/efficiency study, related to the test and basis functions choice, is then presented and discussed. The technique is combined with the use of dynamic ground planes generating mirror basis functions, which completely remove any singular instability. Finally, the computed Green's functions are included into a mixed-potential IE formulation for the accurate and very fast analysis of practical multilayered shielded circuits. The proposed technique does not suffer from any convergence issue and it is extremely competitive in terms of accuracy and efficiency as compared to other methods known to the authors.