This is the first Evidence ever of the Oldest Religion. It is neither a Matriarchy or a Patriarchy. Eanna (Woman) and Anu (Man) ruled side by side, sharing the Uruk each with their own Districts. 

This Union with Man and Woman ruling side by side lasted up until 2,400 BCE.

https://en.wikipedia.org/wiki/Uruk#:~:text=The%20Eanna%20District%20was%20composed,the%20history%20of%20the%20city.

According to the SKL, Uruk was founded by the king Enmerkar. Though the king-list mentions a father before him, the epic Enmerkar and the Lord of Aratta relates that Enmerkar constructed the House of Heaven (Sumerian: e₂-anna; cuneiform: 𒂍𒀭 E₂.AN) for the goddess Inanna in the Eanna District of Uruk. In the Epic of Gilgamesh, Gilgamesh builds the city wall around Uruk and is king of the city.

Uruk went through several phases of growth, from the Early Uruk period (4000–3500 BC) to the Late Uruk period (3500–3100 BC).^[1] The city was formed when two smaller Ubaid settlements merged. The temple complexes at their cores became the Eanna District and the Anu District dedicated to Inanna and Anu, respectively.^[1] The Anu District was originally called ‘Kullaba’ (Kulab or Unug-Kulaba) prior to merging with the Eanna District. Kullaba dates to the Eridu period when it was one of the oldest and most important cities of Sumer.

The Eanna District was composed of several buildings with spaces for workshops, and it was walled off from the city. By contrast, the Anu District was built on a terrace with a temple at the top. It is clear Eanna was dedicated to Inanna from the earliest Uruk period throughout the history of the city.^[12] The rest of the city was composed of typical courtyard houses, grouped by profession of the occupants, in districts around Eanna and Anu. Uruk was extremely well penetrated by a canal system that has been described as “Venice in the desert”.^[13] This canal system flowed throughout the city connecting it with the maritime trade on the ancient Euphrates River as well as the surrounding agricultural belt.