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We present the list of unavoidable local phenomena (bifurcations) occurringon the parabolic and flecnodal curves of evolving smooth surfaces in 3-spaceR^3 (or RP^3). We also present the list of bifurcations occurring on the curveof inflections of the solutions of evolving implicit differential equations(IDE). Our results are based on the properties of the contours of surfaces (ina contact 3-space) for projections all whose fibres are Legendrian. Keywords:Surface, flecnodal curve, contact geometry, implicit differential equations.

Surface Evolution, Implicit Differential Equations and Pairs of Legendrian Fibrations