Find all positive integer solutions verifying two conditions

Let us consider the following Diophantine problem:

Find all positive integer solutions verifying the two conditions:

(1) $$(x+1)^2$$ is a multiple of $$2^{y}$$

(2) $$2^{y}≤x<2^{y+1}$$

where $$y$$ is a fixed positive integer.

Context of the question: Let us consider the following differential equation:

$$z′=f(z,t)$$

where $$f$$ is a continuous function and $$t,z∈ℝ$$

When one search for the number of limit cycles of this equation, the above conditions holds. Now, the problem: Given $$y$$, can we decide whether the above conditions has a solution in positive integers $$x$$. In particular, I am interested on the cases where the number of limit cycles is zero, i.e., the above conditions has no solutions in positive integers $$x$$.

Consider $$x=2^y+2n-1$$, where $$2n-1<2^{y+1}-2^{y}$$, then $$(x+1)^2$$ is a multiple of $$2^y$$ only when $$n=2^{y-1}$$. If $$x$$ is even, then no solutions can exist because it leads to the conclusion that an odd number is a multiple of an even number, i.e., the odd number $$(x+1)^2$$ is a multiple of the even number $$2^{y}$$