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We have explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality alpha between velocity and the curl field of flow is investigated. Conditions for the existence of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered satisfying to Laplace equation. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained. Analytical and numerical findings have been outlined including outstandung graphical presentations of various types of constructed solution in illuminating dynamical snap-shots which demonstrate developing in time the structural behaviour of topology of the aforepresented solutions., Comment: 17 pages, 5 figures; Keywords: couple stress fluid, micropolar fluid, bipolar vector Laplacian, non-stationary helical flow, Beltrami flow