Consider the following system of partial differential equations: 8>< >: Px ¹ + uux + vuy = a(uxx + uyy) + Fx Py ¹ + uvx + vvy = a(vxx + vyy) + Fy ux + vy = 0 (1) P = P(x; y); u = u(x; y); v = v(x; y); x; y 2 R; where P; u; v; F : D ! R2. The system (1) describes the process of stationary fluid flow or gas on a flat surface. The function P represents the pressure of the liquid, and functions u; v represent the flow of the liquid (gas). The constants a > 0 and ¹ > 0 are determined by the parameters of the liquids (of the gas), which are viscosity and liquid’s density. The function F represents the exterior forces. Theorem. Suppose that u; v 2 C2(D) admit the bounded derivatives up to including order 2 in D. If f(z), z = x + iy, is an analytical function in D, then (u; v; P), with u = Imf, v = Ref, P = [F ¡0; 5(u2+v2)+c]¹ are solutions to the system (1). If W(x; y) is a harmonic function in D, then (u; v; P), with u = Wy + c1y + c2; v = ¡Wx + c3x + c4; P = [F ¡ 0; 5(u2 + v2) + (c1 ¡ c3)W + 0; 5(c1y2 ¡ c3x2) + c2y ¡ c4x + c]¹; and the arbitrary constants c; c1; c2; c3; c4 are solutions to the system (1). In addition, various special cases were studied, and particular and exact solutions of the system (1) were found in these cases.