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We expand the classical balayage of measures and subharmonic functions on asystem of rays $S$ with a common origin on the complex plane $\mathbb C$. Thisallows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$build $\delta$-subharmonic function on $\mathbb C$, harmonic outside $S$coinciding with $v$ on $S$ outside the polar set, of the same order of growthas $v$. We give applications to the relationship of the growth of an entirefunction in $S$ with the distribution of its zeros.

Balayage of measures and subharmonic functions on a system of rays. II. Balayage of finite genus and regularity of growth on one ray