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.