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The problem of non-unique solutions at fictitious frequencies that can appear in the boundary element method for external acoustic phenomena described by the Helmholtz equation is studied. We propose a method to fully desingularise in an analytical way the otherwise hyper-singular Burton–Miller framework, where the original boundary element method and its normal derivative are combined. The method considerably simplifies the use of higher-order elements, for example, quadratic curved surface elements. The concept is validated using the example of scattering on a rigid sphere and a rigid cube, and its robustness and effectiveness for external sound-wave problems are confirmed.