Keywords (Extracted from title, table of contents and abstract of thesis)
differential equations, underlying geometry, isometries, lie algebra, autonomous weakly nonlinear systems, spacetimes, geodesic equations 
Abstract In this thesis symmetry methods have been used to solve some differential equations and to find the connection of isometries of some spaces with the symmetries of some related differential (geodesic) equations. It is proved here that the Mc Vittie solution and its nonstatic analogue are the only plane symmetric space times with electromagnetic field. The Einstein equations for non static, shearfree, spherically symmetric, perfect fluid distributions reduce to one second order nonlinear differential equation in the radial parameter. General solution of this equation is obtained in [11] by symmetry analysis. Corrections of some examples of the solution in the earlier work [11], by formulating a general requirement for physical relevance of the solution, are presented. An algebraic proof that the Lie algebra of generators of the system of n differential equations, (yo) = 0, is isomorphic to the Lie algebra of the special linear group of order (n + 2), over the real numbers, is provided. A connection between the symmetries of manifolds and their geodesic equations, which are systems of second order ordinary differential equations, is sought through the geodesic equations of maximally symmetric spaces. Since such spaces have either constant positive, constant negative or zero curvature, three cases are considered. It is proved that for a space admitting so(n + 1) or so(n,l) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so(n + 1)$ d2 or so(n, 1)$ d2 (where d2 is the 2dimensional dilation algebra), while for those admitting so(n) $ J?II the algebra is sl(n + 2). A corresponding result holds on replacing so(n) by so(p, q) with p + q = n. It is conjectured that if the isometry algebra of any underlying space of nonzero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h El:) d2 provided that there is no crosssection of zero curvature. Some results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly nonlinear systems. An adaptation of a theorem that provides the generators that leave the functional involving a Lagrangian for such equations is presented. A detailed example to illustrate the method is given.
