Oblonsky takes full advantage of the pleasures of Petersburg, which he finds liberating in comparison to those of Moscow.
In particular, Oblonsky likes the Petersburg attitude towards money: large debts are not a problem.
Oblonsky pays Princess Betsy a call, and goes a little too far in his flirtations. (In other words, the man either has to make a move or risk insulting Betsy.) Luckily Princess Myagky drops in and inquires after Anna, saying that she's entirely on Anna's side.
Oblonsky tells Princess Myagky of Karenin's response to requests for a divorce, and the Princess tells Oblonsky that Karenin's decision will rest on Jules Landau, a celebrated clairvoyant.
Apparently Countess Lydia and Karenin make all their decisions after consulting him.