Abstract The paper discusses multi-parametric formulations for structural equilibrium problems. By seeking subsets of equilibrium states, which also fulfil extra conditions, more detailed information on structural behaviour can be obtained. These conditions can be related to analyses of, e.g. critical behaviour, imperfections or optimisation. The paper shows a general problem setting and some interesting augmentations. A solution method for the special case of one-dimensional equilibrium subsets is described. The algorithm is a generalisation of a path-following algorithm, where new methods are developed for the evaluation of the path tangent, and for the isolation of special solution points. Examples are used to show the properties and the possibilities of the ideas.