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We consider the effective action of the Heterotic Superstring to first orderin alpha prime and derive the necessary and sufficient conditions that a fieldconfiguration has to satisfy in order to admit at least one Killing spinorusing the spinor bilinear method and making minimal coordinate and framechoices. As a previous step in this derivation, we compute the complete spinorbilinear algebra using the Fierz identities, obtaining as a by-product thealgebra satisfied by the Spin(7) structure contained in the bilinears in anarbitrary basis. We find the relations existing between the left-hand-sides ofthe bosonic equations of motion evaluated on supersymmetric fieldconfigurations using the Killing Spinor Identities instead of the (far morecomplicated) integrability conditions of the Killing Spinor Equations as it iscommon in the literature. We show how to include the Kalb-Ramond's Bianchiidentity in the Killing Spinor Identities.

On the supersymmetric solutions of the Heterotic Superstring effective action