Quaternionic contact structure with integrable complementary distribution

We study positive definite quaternionic contact $(4n+3)$-manifolds ($qc$-manifold for short). Just like the $CR$-structure contains the class of Sasaki manifolds, the $qc$-structure admits a class of $3$-Sasaki manifolds with integrable distribution isomorphic to $\mathfrak{su}(2)$. A big difference concerning the integrable complementary $qc$-distribution $V$ of the $qc$-structure from $3$-Sasaki structure is the existence of Lie algebra not isomorphic to $\mathfrak{su}(2)$. We take up non-compact $qc$-manifolds to find out a salient feature of topology and geometry in case $V$ generates the $qc$-transformations $R^3$.