How to solve the system of such differential equations? Let $x_i=x_i(t)$ be a functions and let consider the system of differential equations
$\left \{
\begin{align} 
&{\frac { d}{{ d}t}}x_{{0}}  =3\,x_{{1}} ,\\
&{\frac { d}{{ d}t}}x_{{1}}  =0,\\
&{\frac { d}{{ d}t}}x_{{2}}  =
{\frac {2\,x_{{1}}  x_{{2}}  +x_{{3}}
  }{x_{{0}}  }},\\
&{\frac { d}{{ d}
t}}x_{{3}}  ={\frac {3\,x_{{1}}  x_{{3
}}  -6\, x_{{2}}   ^{2}
}{x_{{0}}  }}. 
\end{align} \right.
$
My attempt. By brute forse I have found the first integral
$$
{\frac {4\,{x_{{2}}}^{3}+{x_{{3}}}^{2}}{x_0^{2}}}=C,
$$
but I cant find $x_i$ as functions of $t.$ Is it possible?

Maple 2019.1 with "Physics" package version 419:
sol := dsolve([diff(x0(t), t) = 3*x1(t), diff(x1(t), t) = 0, diff(x2(t), t) = (2*x1(t)*x2(t) + x3(t))/x0(t), diff(x3(t), t) = (3*x1(t)*x3(t) - 6*x2(t)^2)/x0(t)])
I have:
[{x1(t) = _C2}, {x0(t) = Int(3*x1(t), t) + _C1}, {diff(x2(t), t, t) = (-6*x1(t)^2*x2(t) + 2*x1(t)*diff(x2(t), t)*x0(t) - 6*x2(t)^2)/x0(t)^2}, {x3(t) = -2*x1(t)*x2(t) + diff(x2(t), t)*x0(t)}] 
If I add build command to dsolve I have a solution:
{x0(t) = 3*_C2*t + _C1, x1(t) = _C2, x2(t) = -9*(_C2*t + _C1/3)^2*WeierstrassP(((3*_C2*t + _C1)^(1/3) + _C3*_C2)/_C2, 0, _C4)/(3*_C2*t + _C1)^(4/3), x3(t) = -(3*_C2*t + _C1)*WeierstrassPPrime(((3*_C2*t + _C1)^(1/3) + _C3*_C2)/_C2, 0, _C4)}
In LaTex code:
$\left\{ {\it x0} \left( t \right) =3\,{\it c2}\,t+{\it c1},{\it x1}
 \left( t \right) ={\it c2},{\it x2} \left( t \right) =-9\,{\frac {
 \left( {\it c2}\,t+{\it c1}/3 \right) ^{2}}{ \left( 3\,{\it c2}\,t+{
\it c1} \right) ^{4/3}}{\it WeierstrassP} \left( {\frac {\sqrt [3]{3\,
{\it c2}\,t+{\it c1}}+{\it c3}\,{\it c2}}{{\it c2}}},0,{\it c4}
 \right) },{\it x3} \left( t \right) =-{\it WeierstrassPPrime} \left( 
{\frac {\sqrt [3]{3\,{\it c2}\,t+{\it c1}}+{\it c3}\,{\it c2}}{{\it c2
}}},0,{\it c4} \right)  \left( 3\,{\it c2}\,t+{\it c1} \right) 
 \right\} 
$
Check by odetest:
odetest(sol, [diff(x0(t), t) = 3*x1(t), diff(x1(t), t) = 0, diff(x2(t), t) = (2*x1(t)*x2(t) + x3(t))/x0(t), diff(x3(t), t) = (3*x1(t)*x3(t) - 6*x2(t)^2)/x0(t)])
[0, 0, 0, 0]
