We study left-invariant foliations $\mathcal{F}$ on semi-Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations $\mathcal{F}$ when the subgroup $K$ is one of the important groups $\mathrm{SU}(2)$, $\mathrm{SL}_2(\mathbb{R})$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$, $\mathrm{SU}(2)\times\mathrm{SL}_2(\mathbb{R})$, $\mathrm{SU}(2)\times\mathrm{SO}(2)$, $\mathrm{SL}_2(\mathbb{R})\times\mathrm{SO}(2)$. This way we construct new multi-dimensional families of Lie groups $G$ carrying such foliations in each case. These foliations $\mathcal{F}$ produce local complex-valued harmonic morphisms on the corresponding Lie group $G$. This means that they provide the existence of solutions to a difficult over-determined non-linear system of partial differential equations.

We study generalized Cauchy-Riemann (GCR)-lightlike submanifolds of indefinite Kaehler manifolds admitting a quarter-symmetric non-metric connection. We derive a condition for a totally umbilical GCR-lightlike submanifold of indefinite Kaehler manifolds admitting a quarter-symmetric non-metric connection to be a totally geodesic submanifold. We study minimal GCR-lightlike submanifolds and obtain characterization theorem for a GCR-lightlike submanifold to be a GCR-lightlike product manifold.

Ikeda and Sakamoto studied a dynamical control problem called the linear first integral for holonomic dynamical systems, and our proposition proved the same result as theirs in integrability. Also, a symplectic Haantjes manifolds has been defined by Tempesta and Tondo, which is a characterization of integrable systems using $(1,1)$ tensor fields. We show integrability in dynamical control problems from a geometric point of view by means of a concrete construction of a symplectic Haantjes manifold.

A plethora of explicit formulas that parameterize any type of the spiric sections are derived from the first principles.