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Every planet or moon has a portal. It is actually not difficult to find them, if you follow these steps exactly:
Make sure you have at least 3 Alien Planetary Charts for Ancient Artifact Sites. Go to your local space station in a normal system (not uncharted or abandoned), and purchase these charts from the Cartographers stall. The cartographers are located on the first floor, to the right of the entrance.
Make sure to have one of the following items, corresponding to the species of the system where you are searching for the portal:
Korvax: Korvax Casing
Gek: Gek Relic
Vy'keen: Vy'keen Dagger
Tip: If you don't have one, the local trade outposts will always sell the counterpart item for the relevant species.
When you are ready, select your planet and land anywhere. It is important to land as the results are erratic if you do this from inside a ship in flight. Exit the ship, use the planetary chart from your inventory. A chart has an equal chance of finding any one of the following:
Ancient ruin
Monolith
Plaque
You want the directions to a monolith, so if you do not get a monolith when you try using a chart, just use another chart. If you get an error message saying "This type has already been found", ignore it, and try again, repeatedly, failed attempts do NOT consume a chart. In the worst case, you will use three charts to finally get directions to the Monolith.
Make your way to the Monolith and complete the test; you must succeed the trial at most monoliths (we recommend you place a save point, save first, so you can reload if you choose wrong). After the test concludes, interact with the monolith again, and it will give the option to locate a portal. Choosing that option will hand over the alien-specific artifact that was mentioned above.
A marker will show up to direct you to the portal.
Important Notes:
Make sure you understand the difference between galactic coordinates and glyphs, if you have galactic coordinates you need to convert them to glyphs to use at a portal. Use the NMS Portal Decoder for this.
In order to use glyphs, you must be in the same galaxy as the region you want to visit. Glyphs cannot bring you to a different galaxy.
Galactic Coordinates (wiki)
Portal Address (wiki)