Canonical $A$-deformations preserving the lengths of lines of curvature on a surface

In this paper, infinitesimal deformations which preserve the area element of a surface in $E_3$ ($A$-deformations) which also preserve the lengths of lines of curvature are studied. Here $A$-deformations are considered up to infinitesimal bendings (which constitute the trivial case for the problem posed). Such $A$-deformations are also called canonical.
For regular surfaces of nonzero total curvature (without umbilic points) the problem indicated reduces to a homogeneous second order partial differential equation of elliptic type. In this paper a series of results about the existence and arbitrariness of canonical $A$-deformations is obtained. The basic results are valid for surfaces in the large.
Bibliography: 20 titles.