In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.PACS numbers: I. INTRODUCTIONNonlinear field theories are ubiquitous in the description of physical systems from particle physics [1] -[4] to condensed matter systems [5] -[7] and cosmology [8], where any genuine interaction is generally related to the nonlinearity of the underlying field theory. In these theories, one powerful strategy to obtain solutions of physical importance is to reduce the order of the original field equations (the Euler-Lagrange (EL) equations) of the system. The resulting equations of lower order -Bogomolnyi equations, self-duality equations, Bäcklund transformations, etc. -are easier to solve and allow to obtain a large number of relevant solutions with particular characteristics, like solitons, nonlinear waves, vortices, monopoles, instantons, etc. There exist several known methods to achieve this reduction of order, where the best-known one is probably the Bogomolnyi trick [9], [10], [11] of completing a square. To consider an example, let us assume that we have the energy functional of a field theory which for static fields may be expressed as a sum of two terms, E = d d x(A 2 + B 2 ) (typically, A depends on first derivatives, whereas B only depends on the fields). This may trivially be rewritten asIf, in addition, Q is a homotopy invariant (i.e., AB is locally a total derivative), then it does not contribute to the EL equations, and its value only depends on the boundary conditions imposed on the fields. As a consequence, E andĒ lead to the same EL equations. Further,Ē is non-negative, so E obeys the inequality E ≥ |Q| (Bogomolnyi bound) which is saturated by solutions to the reduced-order (usually, first-order) equation A = ±B (Bogomolnyi equation or BPS equation).Recently it has been observed [12] that it can be useful to partly invert the logic of this construction. That is to say, let us assume that we have two functionals (functions of the fields, their derivatives, and possibly also of the coordinates x µ ) A, B which are in some sense "duals" of each other, and which are such that the product AB is locally a total derivative (the integral Q = 2 d d xAB is a homotopy invariant). This automatically implies that the "energy functional" E = d d x(A 2 + B 2 ) is a BPS action, and the "self-duality equations" (BPS equations) A = ±B