$\left\{\begin{array}{ccc}\dot{p}_1 & = & \frac{1}{z}p_2p_3\\\dot{p}_2 & = & -\frac{1}{z}p_1p_3\\\dot{p}_3 & = & (\frac{1}{y}-\frac{1}{x})p_1p_2\end{array}\right.$ where $p_1(0)=a,p_2(0)=b,p_3(0)=c$ and $x,y,z$ are constants.
1) how to resolve this system differential equations by hand(not computer)? the fomulas, the calculations...
2) what if $x=y=M$ where $M$ is a constant(has no special means)?
Let's call $1/z=A$ and $1/y-1/x=B$. Multipliying the first equation by $p_1$, the second by $p_2$ and adding we get $$ p_1\dot p_1+p_2\dot p_2=0\implies p_1^2+p_2^2=\text{constant}=a^2+b^2. $$ Multipliying the first equation by $-B\,p_1$, the third by $A\,p_3$ and adding we get $$ -B\,p_1\dot p_1+A\,p_3\dot p_3=0\implies -B\,p_1^2+A\,p_3^2=\text{constant}=-B\,a^2+A\,c^2. $$ We have obtained two first intergrals. We can solve for $p_2$ and $p_3$ in terms of $p_1$ and obtain a first order differential equation for $p_1$.