@verdin 
In the neighborhood of a solution point there will always be a continuous solution (according to the theory). But constraints on variables will have to be chosen from the found set of solutions.
Draghilev's method for your situation is well stated here
https://www.maplesoft.com/applications/view.aspx?SID=149514
If translation from Russian is available to you, then
https://vk.com/doc242471809_437831729

@verdin 
If there are continuous solutions of the system of m underdetermined equations (m*n, n>m), it is possible to obtain the dependence of all variables: xi, i = 1..n (where "a" is one of xi) from a certain parameter. For example, the  Draghilev's method. It is suitable?

@verdin 
I'm afraid you misunderstood. In its original form, your system most likely has no solutions.
It was shown that the separately numerator and denominator (that after "/") have the same solutions. These were equations without denominators in fsolve.

@verdin 
Your system of equations seems to have dependencies and features. In the first text I removed the denominators and found a solution with the help of fsolve. The same values of variables give 0 of denominators (second text).
Please check carefully.
Alg_equation_numerators.mw
Alg_equation_denominators.mw

@verdin 
We are not talking about the Draghilev method, we are talking directly about your system of equations, which needs to be reduced in order to try to get at least some solution - just a point.
I simply showed a way to reduce the system at the expense of the very simple expression of some variables through other variables and the removal of denominators. ( I did not invent anything, but I started with your equations: f1,f2,f3,f4)

@verdin 

You first need to simplify your system of equations. You will have, as a maximum, 5 variables (x4, x5, x6, x7, a) and four equations f5, f6, f7, f8.
Here, for example, the first steps.
Alg_equation_trans_f5.mw

You also need to remove the denominators from the equations, but when you find the solutions, you must to check these solutions to equality 0 denominators.

@rlopez 
Thank you for your attention to the spelling of the last name Draghilev.

Examples:  2x1^2 + x2^2 + x3^2 - 0.5^2 = 0; and x1^4 + x2^4 + x3^4 - 0.5^4 = 0;

 

The algorithm is still being finalized.

As for me, then no, I cannot.  But I can promise, if something turns out interesting, I will definitely try  implement in Maple.

 

@Kitonum 
The highest class of Maple using, Yuri Nikolaevich!
(Maple17 works.)

A reminder of the turbulent inappropriate activity on the forum MaplePrimes
https://www.mapleprimes.com/posts/204297-Is-There-A-Way-To-Avoid-The-Consequences

@spalinowy 
and this is only one variable s. Listen to the first advice by Mariusz Iwaniuk.
( Do not mock your computer.)

@spalinowy 
You said that you are solving a system of 5 equations, right? Which variables do you want to find? Only s? In this case, I'm afraid you do not very well do what you need, and you were given a very correct answer.
If this is not the case, then please let us know in detail what the real situation is.

Will not say  that I understand the problem, but you can build a pyramid with the help of
(Не скажу, что понимаю задачу, но строить пирамиду можно с помощью)
plottools[polygon]
>?plottools[polygon]

@Carl Love 
of course, you suggested a general approach.