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  Purchase 9 books that when completed upgrades your Atma Zodiac Weapon into Animus Zodiac Weapon. Each book costs 500 Allagan Tomestone of Soldiery and is sold by G'jusana at Revenant's Toll in Mor Dhona.
Each book requires you to do the following:
    kill 100 enemies - 10 groups/types of enemies, 10 from each group/types. 
    complete 3 dungeons. 
    complete 3 FATEs. 
    complete 3 battlecraft / Grand Company leves. 
   
※ You must be equipped with the Atma Weapon you wish to enhance when making the purchase.  You can possess only one book in the set of nine at any given time.   Upon completing the objectives in all nine books, present your Atma Weapon to Jalzahn to complete the enhancement process into an Animus. 
The following are the Animus Weapons introduced in patch 2.28:
Lv 50 | 2H Thaumaturge's Arm 
  
Lv 50 | Arcanist's Grimoire 
  
Lv 50 | Arcanist's Grimoire 
  
  
Enemies 
Dungeons 
FATEs 
Leves 
 
  
Enemies 
Raging Harrier
 
Biast
 
Dullahan
 
U'Ghamaro Golem
 
Shoaltooth Sahagin
 
Shelfscale Reaver
 
Sylpheed Sigh
 
Tempered Gladiator
 
Zahar'ak Archer
 
Natalan Boldwing
 
Dungeons 
Brayflox's Longstop
 
The Wanderer's Palace
 
Copperbell Mines (Hard)
 
FATEs 
Breaching South Tidegate
 
Heroes of the 2nd
 
Air Supply
 
Leves 
Don't Forget to Cry
 Rurubana  | Camp Bluefog
  
Yellow Is the New Black
 Lodille  | Whitebrim Front
  
The Museum Is Closed
 Eidhart  | Saint Coinach's Find
  
   
  
Enemies 
Wild Hog
 
Sylpheed Screech
 
Gigas Bonze
 
Zahar'ak Fortuneteller
 
Giant Lugger
 
Hexing Harrier
 
Tempered Orator
 
Shelfclaw Reaver
 
U'Ghamaro Roundsman
 
2nd Cohort Laquerius
 
Dungeons 
Halatali (Hard)
 
Haukke Manor (Hard)
 
The Sunken Temple of Qarn
 
FATEs 
Another Notch on the Torch
 
Everything's Better
 
Return to Cinder
 
Leves 
Circling the Ceruleum
 Rurubana  | Camp Bluefog
  
If You Put It That Way
 Lodille  | Whitebrim Front
  
One Big Problem Solved
 Eidhart  | Saint Coinach's Find
  
   
  
Enemies 
Shelfscale Sahagin
 
Amalj'aa Lancer
 
U'Ghamaro Quarryman
 
Magitek Vanguard
 
Lake Cobra
 
Sylphland Sentinel
 
Mudpuppy
 
Giant Reader
 
Sea Wasp
 
2nd Cohort Eques
 
Dungeons 
Copperbell Mines
 
Dzemael Darkhold
 
Brayflox's Longstop (Hard)
 
FATEs 
Bellyfull
 
The King's Justice
 
Quartz Coupling
 
Leves 
Circling the Ceruleum
 Rurubana  | Camp Bluefog
  
Necrologos: Whispers of the Gem
 Voilinaut  | Whitebrim Front
  
Go Home to Mama
 Eidhart  | Saint Coinach's Find
  
   
  
Enemies 
Iron Tortoise
 
Ked
 
4th Cohort Laquearius
 
Gigas Bhikku
 
U'Ghamaro Bedesman
 
Sylph Bonnet
 
5th Cohort Hoplomachus
 
Shelfeye Reaver
 
Natalan Watchwolf
 
Sapsa Shelfscale
 
Dungeons 
The Thousand Maws of Toto-Rak
 
Amdapor Keep
 
Haukke Manor (Hard)
 
FATEs 
Breaching North Tidegate
 
Black and Nburu
 
Breaking Dawn
 
Leves 
Someone's in the Doghouse
 Rurubana  | Camp Bluefog
  
The Area's a Bit Sketchy
 Voilinaut  | Whitebrim Front
  
Get Off Our Lake
 Eidhart  | Saint Coinach's Find
  
  
  
Enemies 
Amalj'aa Brigand
 
5th Cohort Laquearius
 
Ixali Wildtalon
 
Violet Screech
 
Gigas Sozu
 
U'Ghamaro Priest
 
Lesser Kalong
 
Snow Wolf Pup
 
4th Cohort Secutor
 
Sapsa Shelfclaw
 
Dungeons 
Cutter's Cry
 
Pharos Sirius
 
The Lost City of Amdapor
 
FATEs 
Rude Awakening
 
The Ceruleum Road
 
The Four Winds
 
Leves 
Subduing the Subprime
 Rurubana  | Camp Bluefog
  
Got a Gut Feeling About This
 Voilinaut  | Whitebrim Front
  
Who Writes History
 Eidhart  | Saint Coinach's Find
  
   
  
Enemies 
Sylpheed Snarl
 
5th Cohort Eques
 
Trenchtooth Sahagin
 
Ahriman
 
Sapsa Elbst
 
Natalan Windtalon
 
Amalj'aa Thaumaturge
 
Hippogryph
 
Elite Roundsman
 
2nd Cohort Secutor
 
Dungeons 
Sastasha
 
The Aurum Vale
 
Halatali (Hard)
 
FATEs 
Surprise
 
In Spite of It All
 
Good to be Bud
 
Leves 
Subduing the Subprime
 Rurubana  | Camp Bluefog
  
Someone's Got a Big Mouth
 Lodille  | Whitebrim Front
  
Big, Bad Idea
 K'leytai  | Saint Coinach's Find
  
   
  
Enemies 
Gigas Shramana
 
Dreamtoad
 
4th Cohort Signifier
 
Natalan Fogcaller
 
Watchwolf
 
Zahar'ak Battle Drake
 
5th Cohort Signifier
 
Amalj'aa Archer
 
Sapsa Shelftooth
 
Elite Priest
 
Dungeons 
Haukke Manor
 
Copperbell Mines (Hard)
 
Brayflox's Longstop (Hard)
 
FATEs 
Taken
 
Tower of Power
 
What Gored Before
 
Leves 
Necrologos: Pale Oblation
 Rurubana  | Camp Bluefog
  
The Bloodhounds of Coerthas
 Lodille  | Whitebrim Front
  
Put Your Stomp On It
 K'leytai  | Saint Coinach's Find
  
   
  
Enemies 
Amalj'aa Scavenger
 
Ixali Boldwing
 
Hapalit
 
Violet Sigh
 
Zahar'ak Pugilist
 
Elite Quarryman
 
5th Cohort Secutor
 
Axolotl
 
Natalan Swiftbeak
 
2nd Cohort Signifier
 
Dungeons 
Halatali
 
Amdapor Keep
 
Pharos Sirius
 
FATEs 
The Taste of Fear
 
Schism
 
The Big Bagoly Theory
 
Leves 
Don't Forget to Cry
 Rurubana  | Camp Bluefog
  
No Big Whoop
 Lodille  | Whitebrim Front
  
Necrologos: The Liminal Ones
 K'leytai  | Saint Coinach's Find