Duality between cuspidal butterflies and cuspidal $S_1^-$ singularities on maximal surfaces

We give criteria for cuspidal butterflies and cuspidal $S_1^-$ ($cS_1^-$) singularities in terms of the Weierstrass data for maximal surfaces, and also show non-existence of cuspidal $S_1^+$ ($cS_1^+$) singularities on maximal surfaces. Moreover, we show duality between these singularities considering the conjugate maximal surfaces each other.