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Stellar evolution calculations to the stage of the cooling white dwarf weredone for population~I stars with masses on the main sequence $1M_\odot\leM_0\le 1.5M_\odot$. The final helium flash LTP is shown to occur in post--AGBstars with initial masses $1.3M_\odot\le M_0\le 1.32M_\odot$ for theovershooting parameter $f=0.016$. In the case of more effective overshooting($f=0.018$) the final helium flash occurs at initial masses $1.28M_\odot\leM_0\le 1.3M_\odot$. Fivefold variations of the parameter responsible for themass loss rate during the post--AGB stage do not affect occurrence of the finalhelium flash but lead to perceptible changes of the evolutionary time. Selectedmodels of two evolutionary sequences with initial mass $M_0 = 1.3M_\odot$computed with overshooting parameters $f=0.016$ and $f=0.018$ were used asinitial conditions in solution of the equations of hydrodynamics describingradial oscillations of stars on the stage of the final helium flash ateffective temperatures less than $10^4$ K. The maximum pulsation period$\Pi=117$ day determined for the evolutionary sequence $M_0=1.3M_\odot$,$f=0.016$ is in a good agreement with observational estimates of the period ofFG Sge. The mass, the radius and the effective temperature of the star are$M=0.565M_\odot$, $R=126R_\odot$ and $T_\mathrm{eff}=4445$ K, respectively. Atthe same time the average period change rate of FG Sge from 1960 to 1990 isnearly three time larger than its theoretical estimate.

Radial pulsations of stars on the stage of the final helium flash