Darboux integrability of hyperbolic partial differential equations: is it a property of integrals rather than equations?

We consider functions that depend on x, y, u(x,y) and its derivatives with respect to x. Among them, we are interested in functions g for which there exists an ordinary differential operator such that its composition with the Fr & eacute;chet derivative of g is expressed in terms of x, g and the total derivatives of g with respect to x. Integrals of the smallest orders for Darboux integrable partial differential equations (PDEs) possess the above property (which is completely independent of any PDE). We prove an almost converse statement: if a function g has the above property and there exists an equation for which g is an integral, then this equation admits integrals in the other characteristic and is therefore Darboux integrable. In particular, if g is an integral of the smallest order for a Darboux integrable equation, then any equation admitting the same integral g is Darboux integrable too. These facts can be used to check the already known lists of Darboux integrable equations for completeness and to find new Darboux integrable equations. As an illustration of this approach, we obtain a family of the Darboux integrable equations, which is probably new. We also briefly discuss whether the Darboux integrability can be defined via properties of the integrals only (without employing the PDEs). If this is possible, then the classification of the Darboux integrable equations may be reduced to the classification of the functions with these properties.