Leave Zamiatin's study and make your way down the hall - the next door in line leads to Dr. Olga's study. Open the door and get inside - Kate will be able to eavesdrop on the doctor before she unexpectedly disappears.
Use the opportunity to search the empty office. Begin by approaching the computer found on the desk. Look through the inbox (place the circle in the middle of a message to open it) and read the correspondence with three items: a somebody named Cantin, a detective from United States that looks for Kate, and Dr. Mangoling. Once you've read all the e-mails, look at the bottom left violet field to leave the screen and begin a conversation with the aforementioned detective.
Now, you must go back to Dr. Zamiatin and talk about Dr. Olga again (required for an achievement). Next, you must go the doctor's office again.
Now, approach the handle located between the small drape and an armor rack. Enter a close-up and move the handle. The drape will move away, revealing a shield encrusted with coloured stones. Turn right and take a closer look at the sword found next to the handle.
Arrange a picture by moving the rings on the sword's hilt
Begin by unlocking three rims found on top of the hilt. Remove the pins that hold them in place by pulling them away left and upwards. Now, twist the unlocked rims to rotate the three elements on the bottom side of the hilt in order to create a silhouette seen on the screenshot above. Once you've managed to do that, the blade will be released from the pedestal. Look at its bottom part - it has the same set of coloured stones which you could have seen on the shield. Approach the shield and make a closer look. To proceed further, you'll need to rotate the stones to replicate the pattern seen on the bottom of the sword. The pattern is as follows: blue gem on top, green one on the left, red one below, blue one on the right, and red on the bottom right.
Setting the correct pattern will unlock a secret passage - use it and access an elevator. Press the lift buttons to descend to the basement. Make a few steps ahead - Kate will learn of a conspiracy led by Dr. Olga and Dr. Mangoling.