A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation
\[
d
x
/
d
t
=
A
(
x
)
,
dx/dt = A(x),
\]
with
A
(
0
)
=
0
A(0) = 0
in n variables
x
=
(
x
1
,
…
,
x
n
)
x = ({x_1}, \ldots ,{x_n})
can be reduced to a linear form
\[
d
x
′
d
t
=
∂
A
∂
x
(
0
)
x
′
\frac {{dx’}}{{dt}} = \frac {{\partial A}}{{\partial x}}(0)x’
\]
by a change of variables
x
′
=
f
(
x
)
x’ = f(x)
. A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.