In general relativity, spacetime is described by a mathematical framework known as a pseudo-Riemannian manifold. The symmetries of a pseudo-Riemannian manifold correspond to transformations that leave the geometry of spacetime unchanged. These transformations are described by a group called the isometry group.
However, it is indeed possible to have spacetimes with no isometries, meaning they possess no symmetries. Such spacetimes are known as generic or non-homogeneous spacetimes. In these cases, the geometry of spacetime varies from point to point, and there are no global transformations that can be applied to the entire manifold to leave it invariant.
One example of a non-homogeneous spacetime is a gravitational field produced by a single massive object, such as a black hole or a star. The spacetime around such objects is typically not symmetric, and its geometry depends on the mass, rotation, and other properties of the object.
In contrast, there are also spacetimes that possess a high degree of symmetry, such as the Minkowski spacetime, which is flat and possesses the symmetry of the Poincaré group. Other examples include spacetimes with spherical, cylindrical, or plane symmetry. These symmetric spacetimes allow for certain simplifications in the equations of general relativity and have been extensively studied.
Overall, while there are spacetimes with no symmetries, they are typically more complex and challenging to describe and analyze compared to symmetric spacetimes.