We introduce the notion of a cylindrical bialgebra, which is a
    quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a
    universal solution of a generalized reflection equation, yielding an action of
    cylindrical braid groups on tensor products of its representations. We prove
    that new examples of such universal K-matrices arise from quantum symmetric
    pairs of Kac-Moody type and depend upon the choice of a pair of generalized
    Satake diagrams. In finite type, this yields a refinement of a result obtained
    by Balagovi\’c and Kolb, producing a family of non-equivalent solutions
    interpolating between the quasi-K-matrix originally due to Bao and Wang and the
    full universal K-matrix. Finally, we prove that this construction yields formal
    solutions of the generalized reflection equation with a spectral parameter in
    the case of finite-dimensional representations over the quantum affine algebra

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