We propose an extension of the semi-analytical solutions derived by Lin et al .
( 1965 ) describing the two-dimensional homologous collapse of a self-gravitating rotating cloud having uniform density and spheroidal shape , which includes magnetic field ( with important restrictions ) and thermal pressure .
The evolution of the cloud is reduced to three time dependent ordinary equations allowing to conduct a quick and preliminary investigation of the cloud dynamics during the precollapse phase , for a wide range of parameters .
We apply our model to the collapse of a rotating and magnetized oblate and prolate isothermal core .
Hydrodynamical numerical simulations are performed and comparison with the semi-analytical solutions is discussed .
Under the assumption that all cores are similar , an apparent cloud axis ratio distribution is calculated from the sequence of successive evolutionary states for each of a large set of initial conditions .
The comparison with the observational distribution of the starless dense cores belonging to the catalog of Jijina et al .
( 1999 ) shows a good agreement for the rotating and initially prolate cores ( aspect ratio \simeq 0.5 ) permeated by an helical magnetic field ( \simeq 17 - 20 \mu G for a density of \simeq 10 ^ { 4 } cm ^ { -3 } ) .