The steady and unsteady solutions in the problem of the natural convection of water near its density maximum (about 4 °C) in a rectangle cavity are considered. We use the square with the side length H as one of the kinds of rectangles. The vertical walls of cavity are adiabatic and horizontal walls have an isothermal segment from 0.25H to 0.75H and other adiabatic segments. The temperature on top Tu is less than the temperature on bottom Td, and they are symmetrical relatively to the temperature of density maximum Tinv. The control volume method with SIMPLER algorithm is used in numerical calculations. The steady unsymmetrical, two symmetrical and one unsteady oscillatory water flows may be obtained for fixed Grashof number. In this paper all the solutions are studied in detail. The oscillations of the average Nusselt numbers on top and bottom of the square and the average kinetic energy of water flow are shown for the unsteady solution. The Fourier analysis of oscillatory characteristics is made. The isotherms, streamlines and the distribution of kinetic energy in cavity are shown for all kinds of solutions.