Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad

Let D be a thick dual polar space of rank 3, and let H be a hyperplane of D. Calling the elements of D points, lines and quads, we call a quad aNH singular if H V a 1⁄4 P? V a for some point P, subquadrangular if H V a is a subquadrangle, and ovoidal if H V a is an ovoid. A point P A H of a quad a is said to be deep with respect to a if P? V aHH, and it is called deep if P? HH. We investigate hyperplanes H of D such that no quad is subquadrangular. We generalize a result of Shult proving that, if all quads are singular, then the polar space P 1⁄4 D is an orthogonal polar space Q6ðKÞ for some (not necessarily finite) field K, and the hyperplane H is a split Cayley hexagon HðKÞ. If both singular and ovoidal quads exist, then one of the following holds: 1. H 1⁄4 6 P AO P ? where O is an ovoid of a quad o. 2. There exists one deep point P A H such that all quads containing P are singular and the remaining quads are ovoidal. 3. The set P of deep points with respect to the singular quads is a locally singular hyperplane of a dual polar space D0. D0 is the dual of an orthogonal polar space P0 GQ6ðKÞ for some field K and P0 is a subspace of the dual P of D where the lines of P0 are lines of P. The set P together with the lines of D contained in H form a split Cayley hexagon HðKÞ. The hyperplane H contains all points of D on lines of HðKÞ.