Solutions of nonlinear differential equations tend to have singularities. At singular points, such solutions may not be smooth enough to solve the equation in a classical sense. Allowing singularities may be an important part of a mathematical model. The interplay between the differential equations of the model and the possible singularities is an important topic to understand when interpreting the solutions. For critically nonlinear differential equations, the study of singularities is also interesting with regard to symmetries. The symmetries of the solution may not be the same as those of the equation, and possible symmetries of singularities may be responsible for such a phenomenon. A discrepancy between symmetries can thus complicate the understanding of solutions to an equation. Meanwhile, symmetries can also be useful for finding solutions by reduction to a simpler problem. The article discusses such aspects of the theory and explains the phenomena in a rather simple geometric model. While it is used for illustration only, the models we are actually concerned with are taken from differential geometry and physics.