On new Galilei-invariant equations in two-dimensional spacetime

We study realizations of Galilei groups acting as transformation Lie groups in the space of two independent variables and one dependent variable. Classification of realizations of the Lie algebras AG(1)(1, 1), AG(2)(1, 1), AG(3)(1, 1), A (G) over tilde(1)(1, 1), A (G) over tilde(2)(1, 1) and A (G) over tilde(3)(1, 1) within the class of Lie vector fields is carried out. Utilizing the classification results we have constructed the full sets of second-order scalar differential equations in two-dimensional spacetime invariant under the Lie algebras AG(1)(1, 1) and A (G) over tilde(1)(1, 1).