The integrability of a four-dimensional sixth-order bilinear equation
    associated with the exceptional affine Lie algebra $D_4^{(1)}$ is studied by
    means of the singularity analysis. This equation is shown to pass the
    Painlev\'{e} test in three distinct cases of its coefficients, exactly when the
    equation is effectively a three-dimensional one, equivalent to the BKP

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