The gyraton solutions on generalized Melvin universe with cosmological constant

We present and analyze new exact gyraton solutions of algebraic type II on generalized Melvin universe of type D which admit non-vanishing cosmological constant $\mathrm{\Lambda }$. We show that it generalizes both, gyraton solutions on Melvin and on direct product spacetimes. When we set $\mathrm{\Lambda }=0$ we get solutions on Melvin spacetime and for $\mathrm{\Sigma }=1$ we obtain solutions on direct product spacetimes. We demonstrate that the solutions are member of the Kundt family of spacetimes as its subcases. We show that the Einstein equations reduce to a set of equations on the transverse 2-space. We also discuss the polynomial scalar invariants which are non-constant in general but constant for sub-solutions on direct product spacetimes.