There is a theory to solve systems of linear Diophantine equations over $\mathbb{Z}$, e.g., see here. Then one could select the nonneggative solutions.
In this particualr case, I find it helpful to first solve the system over the field of rational numbers. Then, by linear algebra, the system is equivalent to
\begin{align*}
x_1 & =\frac{b_1 + b_2 - b_3 - b_4 - b_5 + 2x_{10} + x_7 + 2x_8 + x_9}{2}, \\
x_2 & = \frac{b_1 - b_2 + b_3 - b_4 - b_5 + 2x_{10} + 2x_6 + x_7 + x_9}{2},\\
x_3 & = b_4 - x_{10} - x_6 - x_8, \\
x_4 & = b_5 - x_{10} - x_7 - x_9, \\
x_5 & = \frac{- b_1 + b_2 + b_3 + b_4 + b_5 - 2x_{10} - 2x_6 - 3x_7 - 2x_8 - x_9}{2},
\end{align*}
where we can choose the variables on the LHS (namely $x_6,x_7,x_8,x_9,x_{10}$) as arbitrary rational numbers.
Now we can restrict the domain to nonnegative integers, and obtain the additional conditions on $x_6,\ldots ,x_{10}$: the expressions in the nominator for $x_1,x_2,x_5$ need to be even, and all expressions need to be nonnegative.
There is still some work to do, however, but it seems easier to me than before. For example, if $b_4=b_5=0$, then necessarily $x_6=\ldots =x_{10}=0$.
This is more of a hint rather than a solution, but still:
You have $$x_1+x_2+x_3=n.$$ Substituting this into your second equation gives $$x_4+x_5=2n.$$ Substituting both in the third gives $$x_6+x_7+x_8=2n.$$
If I’ve understood correctly you can solve this new system of equations since each $x_i$ appears in exactly $1$ equation.
Best Answer
It is not so easy. The thing with monomials in a function is that it's differential: $$(y(t)^n)' = y'(t)\cdot ny(t)^{n-1}$$
This is product between function and it's derivative. It is non-linear.
I think you need to find some representation where function concatenation with polynomial is linear. This is possible with for example Carleman matrices.
So our function y is represented $\bf M_y$, squaring represented by $\bf M_2$, Differentiation $\bf M_D$
Now we can write $(y(t)^2)'$ as $$\bf M_D M_2M_y$$
But I am not sure that such $\bf M_D$ exists for these Carleman matrices.