The pharaoh with a bird within his serek is known from only one piece of evidence, and that comes from the galleries under the step pyramid from third dynasty king Djoser at Sakkara.
This object is from a stone vessel made of schist with this king's name carved in.
His position is far from certain, but he is considered to have had a short reign at the end of the second dynasty. This estimation is made by comparing his hieroglyphs' and serek's form to those of other rulers at the time. One of them is a crude serek found in king Qaa's tomb at Abydos in 1902 and possibly showing a bird.
This bird looks like a stork with a long body and neck and a rather short nib, possibly a heron and the picture left shows a similar hieroglyph. His position in the sequence of kings during this rather unknown period is hard to establish with certainty.
He must not be confused with the Horus Ba from the third dynasty whose serek had a human bone, in one occasion together with an ram. (The phonetic sounds in Egyptian is similar in these names).
The reason for placing him after Qaa is mainly the epigraphically similarities (the form of the serek) and a seal from his tomb shows the seven first rulers from the first dynasty in a successive line without mentioning a king called "Bird". Accompanying text from the scanty remains of the Bird Pharaoh is almost identical to some of Qaa's, making it very likely to place him just in this era.