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We consider the shape of the free surface of steady pendent rivulets (or equivalently, two-dimensional droplets) beneath a planar substrate. We formulate the governing equations in terms of two closely related dynamical systems and use classical phase-plane techniques, in particular time maps, to analyse the bifurcation structure of the problem. Our results explain why lubrication theory is unable to capture this bifurcation structure for pendent rivulets, although it is successful in the related problem of sessile rivulets. [ABSTRACT FROM AUTHOR] more...