The paper examines the equilibrium stability problem for a simple class of elastic space trusses in the shape of a regular pyramid. Joints located at the vertices of the base polygon are fixed while the joint at the apex is subjected to a proportionally increasing load acting in either the vertical direction, in the horizontal plane, or along a generic oblique direction. Exact closed-form solutions are derived for each load condition under the common hypotheses of linear material law, small or moderate axial deformation in bars and large nodal displacements. Despite their seeming simplicity, these mechanical systems exhibit a wide variety of post-critical responses, not exhausted by the classical snapping and bifurcation phenomena. In addition to regular primary and secondary branches, the equilibrium paths may include neutral branches, namely branches entirely composed of bifurcation or limit points. Besides their immediate theoretical interest, these branches are particularly difficult to handle by the standard numerical procedures of non-linear analysis, so the given solutions may represent severe benchmark tests.