  The Cauchy-Szeg\"o singular integral is a fundamental tool in the study of
holomorphic $H^p$ Hardy space. But for a kind of Siegel domains, the
Cauchy-Szeg\"o kernels are neither product ones nor flag ones on the Shilov
boundaries, which have the structure of nilpotent Lie groups $\mathscr N $ of
step two. We use the lifting method to investigate flag-like singular integrals
on $\mathscr N $, which includes these Cauchy-Szeg\"o ones as a special case.
The lifting group is the product $\tilde {\mathscr N }$ of three Heisenberg
groups, and naturally geometric or analytical objects on $\mathscr N $ are the
projection of those on $\tilde {\mathscr N } $. As in the flag case, we
introduce various notions on $\mathscr N $ adapted to geometric feature of
these kernels, such as tubes, nontangential regions, tube maximal functions,
Littlewood-Paley functions, tents, shards and atoms etc. They have the feature
of tri-parameters, although the second step of the group $\mathscr N$ is only
$2$-dimensional, i.e. there exists a hidden parameter as in the flag case. We
also establish the corresponding Calder\'on reproducing formula,
characterization of $ L ^p (\mathscr N)$ by Littlewood-Paley functions, $ L ^p
$-boundedness of tube maximal functions and flag-like singular integrals and
atomic decomposition of $H^1$ Hardy space on $ {\mathscr N } $.
