Take the ladder and go back to the coaster cart - its active point is at its front, somewhat obscured by the rocket columns (on the left side of the screen). Zoom in on the dashboard and remove the 15 shank - Kate will get back to the place she originally departed from. Exit the cart and talk to Ayawaska and Kurk. Finally, exit the platform with carts (take the stairs near the brick wall - behind the hand rail, to the right).
Exit the amusement park and approach the automaton sitting on the bench right from the stairs. Open its chest and zoom in on the heart and use the activation key - using the mouse, grab it at the end and turn all the way right (achievement: XZ2000...2.0).
Talk to Oscar - the automaton will insist that it's inappropriate for him to move around without "decent attire". You'll have to get some clothes for it.
To do that, go back to the park, take the stairs in the building on the right and reach the platform with carts and, repeating the sequence on the cart's dashboard, get back to Katerina. Enter her room; when Kate excuses the Youkols, enter the room again. Katerina will be sitting at the table on the other side of the room. Talk to her - give the activation key back and tell her you need some clothes.
Kate will get the Key to Andrei's Cabinet. Go to the bedroom and open the cabinet on the right from the bed. Talk to Oscar - he will appear here the very same moment.