  A concept of boundedness of the $\mathbf{L}$-index in joint variables (see in
Bandura A. I., Bordulyak M. T., Skaskiv O. B. "Sufficient conditions of
boundedness of L-index in joint variables", Mat. Stud. 45 (2016), 12--26.
dx.doi.org/10.15330/ms.45.1.12-26) is generalized for analytic in a ball
function. There are proved criteria of boundedness of the $\mathbf{L}$-index in
joint variables which describe local behavior of partial derivatives and
maximum modudus on a skeleton of a polydisc, properties of power series
expansion. Also we obtained analog of Hayman's Theorem.
  As a result, they are applied to study linear higher-order systems of partial
differential equations and to deduce sufficient conditions of boundedness of
the $\mathbf{L}$-index in joint variables for their analytic solutions and to
estimate it growth.
  We used an exhaustion of ball in $\mathbb{C}^n$ by polydiscs. Also growth
estimates of analytic in ball functions of bounded $\mathbf{L}$-index in joint
variables are obtained.
  Note that this paper and a paper "Analytic functions in a bidisc of bounded
L-index in joint variables" (arXiv:1609.04190) do not overlap because analytic
in a ball and analytic in a bidisc functions are different classes of
holomorphic functions. Some results have similar formulations but a bidisc and
a ball are particular geometric objects.
