Symmetry, Integrability and Geometry: Methods and Applications A Two-Component Generalization of the Integrable rdDym Equation ⋆

We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation. Also we find a B\"acklund transformation between our generalization and Bodganov's two-component generalization of the universal hierarchy equation.


Introduction
Recent papers [3,8,16] provide two-component generalizations for the hyper-CR Einstein-Weil structure equation [6,22] s yy = s tx + s y s xx − s x s xy , (1.1) Plebański's second heavenly equation [25] s xz = s ty + s xx s yy − s 2 xy (1.2) and the universal hierarchy equation [18,19,22] s xx = s x s ty − s t s xy . (1.4) s xz = s ty + s xx s yy − s 2 xy + r, r xz = r ty + s yy r xx + s xx r yy − 2s xy r xy , (1.5) and s xx = e r (s x s ty − s t s xy ), respectively, by substituting for r = 0. Other reductions for (1.4) are found in [7,16]: when u = 0, system (1.4) gives the Khokhlov-Zabolotskaya (or dispersionless Kadomtsev-Petviashvili) equation This paper is a contribution to the Special Issue "Geometrical Methods in Mathematical Physics". The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html while substituting for v = u x in (1.4) produces the normal form for the family of equations studied in [7]. Also, we note the reduction v = u y for system (1.4). This reduction yields equation [9,14,17,21].
As it was shown in [3], the reduction s = x for system (1.6) gives the Boyer-Finley equation The purpose of the present paper is to introduce the two-component generalization for equation which is integrable in the following sense: it has the differential covering [2,11,12,13] containing the non-removable parameter λ = 0 [20]. We show that reductions of the generalization include the general r-th dispersionless Dym equation [1] u ty = u x u xy + κu y u xx , which can be obtained by the method of [20]. While the coverings (1.9) and (2.1) are not equivalent w.r.t. the pseudo-group of contact transformations, (2.1) can be derived from (1.9) by the following procedure, see, e.g., [24]. We consider the function p = p(t, x, y) from (1.9) to be defined implicitly by the equation q(t, x, y, p(t, x, y)) = λ with q p = 0. Then for (x 1 , x 2 , x 3 ) = (t, x, y) we have q x i + q p p x i = 0, so p x i = −q x i /q p . Substituting these into (1.9) yields (2.1).
Our main observation in this paper is that the covering (2.1) allows the generalization This system is compatible whenever the two-component system holds. In other words, (2.2) is a covering for system (2.3), (2.4).

Reductions
By the construction, we have the following reduction for system (2. while (2.4) is its differential consequence. The transformation u → −κu maps (3.1) to (1.10). The corresponding reduction of (2.2) produces the covering of (1.10) studied in [20,23].
, we obtain u ty = −u y u xx and its differential consequence. Then we divide this equation by u y , differentiate w.r.t. y and put u y = −e w . This gives the Boyer-Finley equation [4] w ty = (e w ) xx (3.2) This equation is equation (1.7) in a different notation. Substituting for q = e p in the corresponding reduction of (2.2), we have the covering [10,15,26] for equation (3.2): Reduction D. Finally, when we put v = u y − u x into (2.3) and (2.4), we get the equation u ty = u y (u xy − u xx ) and its differential consequence. Then for u y = e w we have the deformed Boyer-Finley equation [5] w ty = (e w ) xy − (e w ) xx , (3.3) and the corresponding reduction of equations (2.2) with q = e s gives the covering for (3.3). This covering in other notations was found in [5,20].

Bäcklund transformations
The substitution maps system (2.2) to system found in [3]. This system is the two-component generalization of the covering of equation (1.3). The compatibility conditions for (4.2) coincide with (1.6). Solving (4.1) for s t , s x , r t , r x yields