einstein exterior; axisymmetric; invariant transformation; asymptotically ﬂat; nlode; arbitrary constant
Using the symmetry reduction approach we have herein examined, under continuous groups of transformations,the invariance of Einstein exterior equations for stationary axisymmetric and rotating case, in conventional and nonconventional forms, that is a coupled system of nonlinear partial differential equations of second order. More speciﬁcally, the said technique yields the invariant transformation that reduces the given system of partial differential equations to a system of nonlinear ordinary differential equations (nlodes) which, in the case of conventional form, is reduced to a single nlode of second order. The ﬁrst integral of the resulting nlode has been obtained via invariant-variational principle and Noether's theorem and involves an integration constant. Depending upon the choice of the arbitrary constant two different forms of the exact solutions are indicated. The generalized forms of Weyl and Schwarzschild solutions for the case of no spin have also been deduced as particular cases. Investigation of nonconventional form of Einstein exterior equations has not only led to the recovery of solutions obtained through conventional
form but it also yields physically important asymptotically ﬂat solutions. In a particular case, a single third
order nlode has been derived which evidently opens up the possibility of ﬁnding many further interesting
solutions of the exterior ﬁeld equations.