  Rota--Baxter operators over groups have been recently defined in
\cite{LHY2021}, and they share a close connection with skew braces, as
demonstrated in \cite{VV2022}. In this paper, we classify all Rota--Baxter
operators of weight 1 over the Heisenberg Lie algebra of dimension 3 by
directly solving the operators defining equations. Using the fact that the
exponential map from the Heisenberg Lie algebra to the Heisenberg Group is
bijective, we induces these operators to the Heisenberg Group. Finally, we
enumerate all skew left brace structures over the Heisenberg Group induced by
these Rota--Baxter operators.
