Singular plane sections and the conics in the Fermat quintic threefold

We present explicit equations for the space of conics in the Fermat quintic threefold X, working within the space of plane sections of X with two singular marked points. This space of two-pointed singular plane sections has a birational morphism to the space of bitangent lines to the Fermat quintic threefold, which in its turn is birational to a 625-to-1 cover of P⁴. We illustrate the use of the resulting equations in identifying special cases of one-dimensional families of conics in X.