Everything depends on the meaning of the word "analytical".

In the common usage there is a word "analytic", which means "can be expanded in a convergent power series". Then there is a general theorem: if you have
a differential equation $y'=F(y)$, where $y$ is a function with values in $R^n$, and $F$ is an analytic function
(in your case, $F$ is a polynomial of degree $2$), then for every $t_0$ and every $y_0$,
there exists a solution with the property $y(t_0)=y_0$, which is analytic in some
neighborhood of $t_0$. The neighborhood depends on $F$ and on $y_0$.

However I suspect that your word "analytical" means some kind of "explicit formula".
Of course, this has to be defined exactly, but if you are looking for solutions in terms
of functions that contain algebraic functions, exponentials, logarithms, any sums, products, ratios or superpositions of the above and integrals of the above, then the answer is no.

Example:
$$y_1'=y_1^2+y_2,\quad y_2^\prime=1.$$
This is equivalent to the single equation $y'=y^2+t+C,$ and J. Liouville proved in
XIX century that it has no solutions of the type described above.