Examples of two-dimensional pro-$p$-groups and Lie algebras

Several examples of pro-$p$-groups and profinite-nilpotent Lie algebras of dimension two are given. In particular, a negative answer is given to one of the questions raised by L.V. Kuz'min. An essential role in the construction of these examples is played by the author's criterion regarding dimension two and by H. Koch's results.