Mathematics & Physics 2015, 8(3), 320–326 УДК 517.95 + 532 A Hydrostatic Model for an Ideal Fluid: Group Properties of Equations and their Solutions Alexander A.Rodionov∗ Institute of Computational Modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Russia Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 07.01.2015, received in revised form 24.02.2015, accepted 27.05.2015 Group properties of hydrostatic model equations of a layer motion in an ideal fluid on a function defining the free surface and the thickness of the fluid layer under the free boundary are studied. <...> Examples of several exact solutions in Cartesian and cylindrical coordinates are given, they determine the free surface and the pressure on it. <...> Basic equations Consider equations of motion for an ideal incompressible fluid in a gravitational field ut +uux +vuy +wuz + 1 ρpx = 0, vt +uvx +vvy +wvz + 1 wt +uwx +vwy +wwz + 1 ρpz = −g, ux +vy +wz = 0. ρpy = 0, (1) Here u, v, w are components of the velocity vector; the pressure p is the function of the variables x, y, z and of time t; the fluid density ρ is constant (we can take ρ = 1); g = const > 0 is the acceleration of the force of gravity which acts in the negative direction of the z axis. <...> In this situation the system (1) is rewritten in the following form (4) Let z = η(x, y, t) be the equation of the free boundary on which the dynamic and kinematic conditions are fulfilled (5) Alexander A.Rodionov A Hydrostatic Model for an Ideal Fluid: Group Properties of Equations . ηt +u(x, y, η(x, y, t), t)ηx +v(x, y, η(x, y, t), t)ηy = w(x, y, η(x, y, t), t), (6) where pa(x, y, t) is the atmospheric pressure. <...> From formula (3) taking into consideration the condition (5) on the free surface we find that pa(x, y, t) = −gη(x, y, t)+q(x, y <...>