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The boundary integral equation is solved analytically in the case of two- and three-dimensional growth of angled dendrites and arbitrary parabolic/paraboloidal solid/liquid interfaces. The undercooling of a binary melt and the solute concentration at the phase transition boundary are found. The theory under consideration has a potential impact in describing more complex growth shapes and interfaces. © 2020 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.